Firstly, I'd try explaining to your DM how you are struggling atm and ask her to have dd for x 2 hours this weekend.
Second, I'd promise yourself x 5 days off sick as soon as dd goes back to school. You can self-certify the first 7 days of sickness yourself, by law.
In that week research charities that might be able to help. Definitely include Home Start voluntary buddies in that. But also have a plan for each day that puts you first.
I agree with banning your ex from your house. It's not fair and not right that you don't get that break and have your home and space invaded by someone who is hostile to you.
You don't mention your finances as an issue, so if you can manage say £30 a month to pay a nice responsible teen to take your dd eg swimming that will be some time to look forward to.
Also, if your dd if mild SEN could you start working on getting her to accept you can go the bathroom for say a 20 min bath on your own while she watches tv. Basically you need to train her into giving you a few mini breaks through the day, for the sake of both of you.
Does she have friends? I found when a friend was round I got a great break.
Lastly, if you ask for a carer's assessment then a social worker can plan temp fostering for when you are in hospital. It's important that you look after your health and they should be able to help you do that.
You have my utmost sympathy and sound at the end of your tether. Hang on though, nothing stays the same forever. Look after yourself x