ASSUME 40% CDR and rank 5 in the healing skill:

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Soraka heals at a 60% AP ratio every 1.2s.
Yuumi heals at a 40% AP ratio every 4.8s.
Kayle heals at a 30%+30% AP ratio every 9s.
Bard can place a new 60% AP ratio shrine every 8.4s.
Nidalee heals at a 32.5%-65% AP ratio every 7.2s.

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Here is the "healing per second" for each of them:

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50% AP -- Soraka
8.3% AP - Yuumi
6.7% AP - Kayle (3.33%+3.33%)
7.1% AP - Bard (~3% if using a direct heal)
~9% AP -- Nidalee (~4.6% if barely hurt)

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Kayle, Bard, and Nidalee are all comparable healers to Yuumi. They are not quite as strong at rapidly healing someone up, but they can get in more immediate healing that Yuumi. Bard, with some well placed Shrines, can get in a 210% ratio burst of healing. Kayle and Nidalee can both get 60% bursts. Yuumi can't. She has to wait 4.8s to get to an 80% burst, and fights are sometimes over by then. But consider how extended fight healing plays out over 10s:

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Yuumi heals 3x; 120% burst.
Kayle heals twice; 120% burst.
Bard heals twice (plus 3 Shrines?); 240% burst. (120% w/ only 1 shrine)
Nidalee heals twice; ~120% burst.

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Now look at the offense during that same 10s:

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An attached Yuumi will launch Q three times, and maybe she will ult.
Kayle will deal enough damage to kill 2-3 champions.
Bard will deal enough damage to kill 2-3 champions.
Nidalee will deal enough damage to kill 2-3 champions.

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See the issue? Yuumi isn't outperforming other healers in long fights, is underperforming relative to them them in short fights, and is doing thousands less offensively in damage than other healers do. Yuumi, as a healer, isn't the bees knees people like to pretend she is. Basically... AP Yuumi sucks.

Yuumi's strength is not that she can heal... it's that she can attach. She can ride along on a teammate, relatively safe from harm, until the right moment to POUNCE. She can heal once, pop off to assist with damage, then hop back on to heal again after the kill (or heal herself then hop on, if the enemy changed targets).

I have two builds that work with this idea of maximizing her attach mechanic rather than her lackluster healing prowess. One is a nearly full offense "crit Yuumi" that also uses Hail of Blades to front-load damage; Ionian Boots, Runaan's Hurricane, Essence Reaver, Infinity Edge, Wit's End, and Guardian Angel. The other build is this one; a tank/ADC/healer hybrid that can be the initiator for your team when called upon. Both can carry, but this build does it with less chance of dying while "crit Yuumi" is far more potent but a bit reliant on not facing fed assassins (which are way too common these days).

Also, are you aware that Yuumi's base AD is 55? 55.

Go back a chapter, and note that during extended fights Yuumi might heal three times to the two times of every other champion. This means the base value of healing is a small edge for her over other healers --- basically she heals for at least 150 more health than other healers over that 10s moment in the game, relative to other healers. AP may be bad on Yuumi, but Mikael's Crucible's +20% to heals/shields is not. That's an extra 42 more healing per heal from Yuumi (and also 12-80 hp per passive shield as a perk). Basically, Yuumi is still a healer; you do still want some healing ramp if you can fit it into even a tank build, which thankfully you can since Mikael's Crucible grants MR. It's the best healing item for her (imo).

Yuumi's rank 9 heal, with Mikael's Crucible on her and Spirit Visage on her host, restores 325 health (per E used). That's an extra 345 health over the 10s window in those extended fights, enough for your host to absorb another blow from anything that doesn't turbo in the damage so fast that healing is irrelevant. (With just Mikael's Crucible, you get an extra 126 health per E in those 10s; this is why Spirit Visage on your target is good.)

Lastly, you might in extremely long games want to convert into an AP Yuumi. Gathering Storm will have stacked up, and it has a 1:1 ratio when converting to AP. You can get rid of BoTRK and Frozen Mallet, build Nashtor's Tooth and Rabadon's Deathcap, then have all that AP ramped by Mikael's Crucible's +20% healing. It's not better than what other healers can do, mind you. But once you have enough AP to make Nashtor's Tooth a serious weapon, APC/tank Yuumi isn't that terrible. You just can't start the game that way, because AP Yuumi builds do virtually no damage for the first 30 minutes of a game.

Yuumi's Q sucks. This means her auto-attacks are keystone to being any kind of offensive threat. You want to get your offense high enough to be capable of killing a target that isn't killing you, so that you can at least push people off of your teammates instead of being ignored in fights.

The fastest item to ramp offense is Blade of the Ruined Kin. It ramps dps faster than a crit item like Infinity Edge or Essence Reaver, and you also gain an active slow to help kill targets. If it is safe to do so, spend 100% of your money toward this item until it is completed -- the more likely scenario is that you will spend 300g on basic boots before you get BotRK finished, and you might rarely want to buy a 300g cloth armor.

After that, ideally you want to keep ramping BotRK. Berserker's Greaves, a Recurve Bow, and a Dagger are the next buys. You don't need to complete Wit's End right away, though it does boost your offense even further (by 13% AS and an ever-increasing amount of on-hit damage). It also increases your move speed by 5%; handy. I almost always go for Wit's End quickly even against physical damage opponents, unless I am having trouble fighting them. If you need to get armor before completing Wit's End, it can be good to get Warden's Mail since it protects you quite a lot for just 1000g.

In almost every game, you are looking for this set-up by mid-game:
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Berserker's Greaves
Blade of the Ruined King
Wit's End
Warden's Mail
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Short games will be ending soon after you get to this point. But with long games, you have a choice to make based on the game state. The main variable to check is who on the enemy team can easily kill you, or at least who on the enemy team is having an easy time killing your teammates. You want to counter-build so that when you confront them, you can at least badly hurt them even if you can't hope to kill them. But sometimes there isn't anyone scary, or there are too many and you can't counter them all at once.

MURDER IS ALWAYS A GOOD OPTION
If you want to be stronger at helping to secure kills or at solo-killing, build Frozen Mallet right away. Enemy players who are fed have huge bounties on them; you want to build this to help your team collect and catch up. If you are winning, this becomes instead a way to help wipe the enemy team then close the game (or at least get an objective). It's never a bad call... just not always the right call.

SUPPORT YUUMI IS BACK ON THE MENU
Sometimes you have a fed teammate that you want to protect, or your team's only strong engage tool for team fights is Yuumi's ultimate. Instead of spending 3100g on Frozen Mallet, spend that gold on finishing Frozen Heart and getting a Forbidden Idol. This gets you to 40% CDR to ult and heal more often, gives your fed teammate a handy defensive aura, and ramps your healng a bit. Once you commit to these items, it's good complete Mikael's Crucible afterward so that you can free your fed teammate from any cc and also heal them for even more.

WHEN SHOULD YOUR TEAMMATES BUILD SPIRIT VISAGE?
Most champions have 2-3 core items that they "need" to function. Typically your teammates should not disrupt this part of their build, although it can be good to get Spirit Visage third even then sometimes. But when it comes time to choose an MR or health item? If they don't pick Spirit Visage, they don't want healed by Yuumi. They are sending a clear message that they think Yuumi should be using her time and mana to heal a different teammate. Not every teammate needs to buy one; only a teammate who has become fed and has tons of gold really needs one. Between Mikael's Crucible on Yuumi and Spirit Visage on her carry, Yuumi's healing is easily more then sufficient to keep the fed teammate alive.

WHAT IF THEY DON'T BUY SPIRIT VISAGE?
Sometimes, no teammate will buy one. It might not even be the right call. This is usually because you have become the carry. You have been getting kills/assists, you are key to winning fights, and you are in hunting mode. Instead of Yuumi guarding her team, her team is guarding Yuumi. You engage, then you jump to a teammate if you get into danger. After that you let your heal reset and hop off again. The enemy team has to deal with a reasonably high octane self-healing tank that can vanish if they don't burn cc on it. You can sometimes see these games coming. You have 10 kills and 10 assists before 20 minutes. You haven't died yet, or maybe only once/twice. You got a lot of tower plates in laning phase. The biggest sign is when you feel you can 1v1 anyone on the enemy side with fair chances... which happens a lot more often than you might think.

SOLO-KILLING OBJECTIVES
While trying this sneaky play is not always the right call, Yuumi with Berseker's Greaves + Blade of the Ruined King can EASILY solo a dragon or the rift herald by herself. Nobody expects the Yuumi to go there, even if they don't see her on the map. You can end the fight at pretty much full health since you heal yourself, with enough mana left to fight/escape someone on your way out. If there is a plant nearby, you are good to go. If you like this sort of cheeky play, go for it. The moment you have BotRK and some attack speed Yuumi has no trouble at all with objectives. BotRK alone can do it if you think you have enough time and won't be discovered, but that's a bit dicey since the fight is a lot longer and you can be out of resources by the end.


SOME RUNE CHOICES ARE KEYSTONE
Hail of Blades, when playing as support, is a huge deal. This is how Yuumi procs Spectral Sickle quickly for gold, so that she can rush Blade of the Ruined King. It also front-loads 3-4 rapid attacks BotRK on-hit to get an enemy champion low, so that your ADC can get an easy kill on the (probably slowed from your BotRK's active) target. The same rune when playing top/mid is part of how Yuumi short-trades effectively with enemy champions that could otherwise shrug off her poke damage.

Ultimate Hunter, due to how much team fighting, kill hunting, and gank escaping power is in Yuumi's ultimate... is essential. You want Yuumi's ultimate up as often as possible. The only reasonable alternative here is Ingenious Hunter to shorten the BotRK and Mikael's Crucible actives -- I didn't like it since Mikael's Crucible is sometimes my last item, but it's not bad.

Gathering Storm is to help Yuumi continue to grow as a threat in long games. But it can 100% be replaced if you want an early game rune instead. Manaflow Band is the other way to go I think. You really want Transcendence, and those are the two strongest runes for Yuumi other than Transcendence. Nimbus Cloak isn't needed because Yuumi is already fast and will be building slows. Scorch is terrible on her due to its long cooldown; it goes better on bursty champs.

YUUMI HAS "INFINITE" SUSTAIN; WHEN TO RECALL
First, pay more attention to your gold than your health/mana. Yuumi will pretty much never be forced out of lane, unless the enemy can tower dive you while you are low. After level 6, even that can be dicey. You can shield yourself, heal, run around really fast, and root them under your tower while you do it. Yuumi isn't that easy to tower dive, especially if your Flash is up too. But gold? Gold is important as a recall marker for her.

Yuumi scales much more off of items than off of levels. This means that when you have the gold to ramp your offense, take whatever window you have to recall and get something. The right order to buy depends on what you are trying to do; deal damage or not die. If you are trying to deal damage, get a Recurve Bow and then 1-2 Longswords -- complete the BotRK before getting Berserker's Greaves, which will net you at least 2000 damage by the twenty minute mark over getting full boots first. If you are needing to stay alive in a tricky lane, get only the Recurve Bow and get Berserker's Greaves before getting any Longswords. If you are getting your butt handed to you, buy Berserker's Greaves first and you'll probably want to get your armor (1000g Warden's Mail) or MR (720g Negatron Cloak) early to help defend yourself against the problem enemy champion. That's a worst case scenario, and you want to try and avoid needing to do it if you can.

The second you have 1000g for Recurve Bow, look to make your first recall. After that, recalls are easy for a bit. You can get Longswords, work toward Beserker's Greaves, or if you get a lot of plates and/or a juicy kill or two you might even get BotRK early. When you can get BotRK, Wit's End, or Berserker's Greaves... don't hang around trying to get something done. Get out of there, gear up, and come back far more capable of getting something done. With Yuumi, it is easy to miss out on thousands of damage (by game's end) by sitting on your gold.

Once you have the mid-game core, you can start basing less frequently. You have almost all of your offense, and you have +40 armor +50 MR. If building to Frozen Mallet, I sometimes will go grab the health if I don't see anything good to do. But there are games where I'm out so long that I just buy the whole item at once. It can be good to hold onto gold at this point until you are sure what you'll be buying next.

GAMES WHERE FROZEN MALLET ISN'T FROZEN MALLET
When playing as support, you don't ALWAYS do without it. In some games, you'll build it anyway then sell your support item if you want to get Mikael's Crucible as a sixth item. But often, as support you'll want to defend your ADC rather than get Frozen Mallet; BotRK's active and your ultimate can be enough to get your ADC their needed kills, so you'll have a high value teammate to defend.

When playing top/mid though, things get more complex:

Some enemies, like Ziggs and Illaoi, can push into your tower stupidly hard. You have to get BotRK to fight them at all, but after that you want Runaan's Hurricane to match their waveclear speed at least somewhat so that you can defend your tower. This item slot is almost always going to be Frozen Mallet's slot, although in some games it can be Mikael's Crucible. You need to look not just at who you are against. Look at how they are playing, and also look at your jungler to see if they are helping you keep pressure off your tower. You must make sure you haven't spent your Runaan's Hurricane gold on some other item already. The only way to do it is to take a look at these variable just as BotRK completes.

A particular note on Runaan's Hurricane; this is an item you gen as mid/top, so healing others isn't as important as healing yourself. Since you are likely giving up your Frozen Mallet, it can be reasonable to get Spirit Visage instead of Mikael's Crucible so that you are getting a bit of health to go with your armor/MR. The price for the 450 health is pretty good. I think that Mikael's is likely still better since it helps your teammates too, but every game is different and it's good to have options.

In other games, BotRK + Wit's End aren't quite enough to dent a tanky fed enemy. You want more offense. Runaan's Hurricane, thanks to the 25% crit chance, isn't that bad of a choice and also punishes the enemy's teammates for grouping too tightly with the fed target you want to bring down. But sometimes you have to break out the machine gun; Guinsoo's Rageblade, which is FANTASTIC with Hail of Blades to get it ramped up fast every time you engage. Typically this will replace the Frozen Mallet, although again in some games it will replace Mikael's Crucible. (It's a matter of which one of those you need more in the late game. Mallet is for health and slow; Crucible is for healing, MR, and sometimes the anti-cc active.)

LoL is far too expansive to list which champions this Yuumi build "beats" or "doesn't beat". Sometimes you are a level ahead, with half an item more than someone. Sometimes it's the other way around. You have to feel it out; use your general experience playing LoL. When in doubt, I suggest building Frozen Mallet and hoping for the best; at the very least, it is the best item you can build (other than BotRK) to help your team secure kills. It has a whopping 700 health on it too, so you are defended from all kinds of damage. It's easily the best choice to stay alive 1v1 vs any bursty champion like an assassin or mage, and it kites some melee champions (like Sett) so hard that you can kill them and still be at full health afterward.

Play this Yuumi build like semi-tank Bard. You have ranged attacks that hurt, you heal and take reduced damage, you have some pursuit/escape buttons, and you are weak early but get much stronger as you get more items. Bard is effectively faster around the map thanks to chimes and magical journey, he has more cc weapons, and he hits a bit harder when he has meeps. Yuumi is effectively tankier, can jump on teammates to help pursue or to save herself, and she has a better (but very similar in purpose) ultimate.

Last of all, be careful before level 5-6. Until you have 3 ranks in your heal, you aren't able to heal yourself well (or quickly) if you take damage. And until you have your ultimate, you don't have a good way to feed your jungler a kill (although it's not as much of a problem if they have their own cc) and you don't have an anti-ganking tool to stop the enemy jungler. Especially at level 1, Yuumi is vulnerable to burst-down cheeses. Caitlyn, Graves, and a number of other champions who are strong early can get lucky if you are careless before you hit level 2.

Please give this build a try or two, then come back and vote. I'm confident that you and everyone on your team who didn't AFK or rage-quit will be pleasantly surprised at what Yuumi can do as an ADC/tank/support ground fighter. It is pretty easy to carry games on her, as long as you focus on winning rather than getting kills. I'd like to get at least 6 votes and the positive rating that I know this build deserves, so that Yuumi players can have way more fun than they currently are.