OK, so let's see if I've got this:
You currently have DSD half the week, so reluctant to get babysitters as that would eat into your limited time with her. (When does she go to bed? Couldn't you just go out later?)
In theory mum has her half the week. In fact she's palming her off on randoms when she has her particularly Friday and Saturday nights.
You have several other concerns about her parenting. Some not really enough to do anything about - sandwiches for tea isn't brilliant, but if she's having school lunches it's not that bad either. You think her house is kept too cold and it's contributing to illness (evidence?). Sharing a bed with new boyfriend sounds like the clearest concern to me.
You don't fancy offering to swap days so you always have weekends, and that would only improve the babysitting aspect.
What I'd do:
Start logging stuff. Log who's looked after her when. Log when she reports sleeping in bed with the boyfriend. Log when she's ill and if she says she was cold. Get ducks in a row so that either she will be encouraged to sort things out a bit, or you will be in a good position to get arrangements changed to give you full or near full residence.
I would email mum to say that you don't think bed sharing with boyfriend is appropriate, and also email concern each and every time DSD is babysat by someone you don't consider appropriate. So it's in writing, and she may respond in writing too. If she emails back that she thinks bed sharing is OK, you could raise that as a concern with someone professional e.g. school nursing service or HV team responsible for 5-18, even Social Services or GP and see what they think.