Agree with others that lack of sleep is a good part of why you're so tired.
Skeleton, I have to say that I was initially a bit put out at your first post ... but then read on, and realised that the times when I've struggled to cope with the days have been when DD is waking lots during the night, and I can't do anything but crash during those daytime naps!
OP, when you've had a poor sleeper then waking 'only a few times a night' seems great in comparison, and you can kid yourself that it's perfectly normal and manageable. But it isn't - not long term.
I used to do all the nights too, but when I went back to work, my DH started doing half the night on my work days and at weekends: we switch over at about 3am. DH usually stays up until about midnight anyway, and I go to bed at 9pm. That means that I sometimes don't have to get up at all in the night (you do wake up briefly, but knowing someone else is dealing with it means you can go straight back to sleep). A few nights of blissful sleep, and you realise quite how much those night wakings are killing you. I really can't emphasize enough how much of a difference that help makes.
Could you do the same, and have your DH do some of the night wakings? If DD just needs a tuck-in, then even 1950s man should be able to cope 
. And as a side effect, he'll realise how hard the nights are, and also stop seeing it as 'your' problem - both of which will make him keener to help. That's what I found anyway!
Also, you say you're knackered by 9pm - I don't know whether you actually go to bed then, but depressing as it is, that might help. I find that even with help it isn't really an uninterrupted night and I genuinely need 9pm-5/6am (which is when my DD usually wakes), especially since I don't get a chance to re-charge at weekends like I did pre-baby. DH is more of a lie-in person than me, so I usually let him have the weekend lie-ins unless I'm really shattered.
If you can possibly afford it, do also get a cleaner. A professional who can just focus on cleaning gets a lot done in 2-3 hours a week, and then with the hard stuff done, you can just do a little bit of maintenance through the rest of the week. And having that time and energy for DD is probably worth more than doing something else expensive with her.
I don't know anything about your health conditions - except that I those things can increase tiredness! Looking after a young child is hard enough when you're in good health. It must be really hard with that extra burden.
Do make the changes you need to, in order to get your energy back. You owe it to yourself and also your daughter. Good luck!