Firstly, his health vs his ability to fight as a captain. This sparked off a lot of things for me. Though Kaien told Rukia he pretty much runs the squad because of Ukitake's weakness, if that was the case, now he's gone Ukitake would have to have a Vice Captain in position in order to just get duties taken care of. Which must mean that Ukitake's health isn't allowed to interfere as much as sometimes it's implied it is.
(Do we know officially what age - roughly - he contracted TB, by the way? I know it was at a young age, before he was involved in Yamajii's academy, but not beyond that).
He was also given the role of Captain even though he had been diagnosed with TB long before that and was obviously not physically fit from time to time. Yamajii says quite clearly that Ukitake can fight for a long period and find the strength if he needed to...and that's my impression too.
Obviously the illness is always underlying and there and it is a factor in things in his every day life.
The moments we see Ukitake MOST weakened by his illness are often times when another type of strength is called into the equation. It's not so much fighting and using his reiryoku that makes him sick, but emotional strain of one sort or another. This seems particularly clear in the following scenes:
~When Kiyone and Sentarou wake him and tell him of Aizen's murder
~Worrying about the fact his advanced expeditionary force have not reported back within the timescale he expected (ie, Kaien's wife, etc)
~Having to fight Kaien and knowing that he'll be killing him if he does.
~Yelling at Byakuya about the fact Rukia's execution is the next day and he's doing nothing about it.
All these events bring on fits of coughing - the third one to the point of blood. In both those moments, he's stopped in his tracks by his illness. Yet at other times when he's shown as being ill, he's managed to push it aside and carry on (like when he first goes out to Soukyoku Hill to interfere between Byakuya and the ryoka).
So I think Ukitake's main weakness, and the biggest trigger to his illness is the emotional factor - it's rooted in what is probably his greatest strength. He bonds with and takes very particular interest in every one of the people who fight for him. Losing Kaien and the other officers at that time must have been devastating to his physical condition.
Which may also explain why he has not appointed a Vice Captain or been required to do so by Yamajii. I think that Ukitake probably selects his Vice Captain on who he feels he can work most closely with and trust the most, therefore it's a position which he'd rather leave vacant than fill for the sake of it.
Which then begs the question, if there was to be another Vice Captain of 13-Bantai - leaving her involvement with the real world aside for the moment - what potential does a Shinigami like Rukia have to fill that position?
Right now, Byakuya is the only thing standing between her and a seated ranking in 13-Bantai. She's just defeated an Arrancar of the Espada on her own, despite the risk to her own health in the process. Some of the Captains have struggled to do that in similar situations. If Byakuya was to be persuaded to release his interdict, could Rukia potentially be Vice Captain of 13-Bantai at some point in the future...?
Ukitake certainly has gone to great lengths to help and protect Rukia since Byakuya entrusted her to his squad, even if it hasn't always gone right. Yet Byakuya seems to still trust Ukitake, and treats him in a way he doesn't treat other people. He doesn't object to being called 'Byakuya' by Ukitake, and refers to him more or less respectfully (as opposed to "kisama" to Ichigo or Yoruichi when doing the same thing). It's my impression that though Ukitake's family ranks below Byakuya's, Ukitake's probably had some involvement with the Kuchiki-ke for a long time. He may even have been involved in some way in helping Byakuya to locate Rukia - or to marry Hisana, or in any of these things which mean that Byakuya is slightly less icily cold to Ukitake than he is to most of the other Captains.
Given that, he might well be persuaded that Rukia is a suitable candidate for 13-Bantai Vice Captain one day...?
Thoughts please? :)