Right here's some practical stuff too - sorry it's really long, I am feeling for you and got carried away!
I did a 'book of everything' with all the house stuff, routines, job description, childrens do/ dont s etc in. Went through it over a coffee & really discussed it, made changes as I went so the nanny felt involved & listened to.... Then revisited the month after to reiterate stuff & problem solve. I thought I was being incredibly anal but it does work as long as the 'book' is kept up to date & you make sure it doesn't feel tooo prescriptive. I should say I am very ill so it's vital things get done a certain way, but it sounds like you need it too!
The 'book of everything' (actually a pretty folder) made me do alot if thinking over how I needed things to be done, & what assumptions I make about my household... Which is really helpful in making it easier to manage the nanny/ cleaner etc. I was worried about being too authoritarian/ controlling, but it seems not to scare people away, the nannys I've had like knowing things are written down & helpful.
Made me realise lots of assumptions I have though! Like, I want my child to help with cooking/ housework, as I want him to learn that these things don't magically happen... Nanny expressed surprise as previously she had to do housework stuff when the kids were napping.
I also want Ds to help tidy, be polite etc, so having a list of what he should be doing & how she should involve him is key - may sound obvious to you but all families differ so good to be clear.
Also I'm like you, can't afford endless 'entertainment' & don't think it's necessary either. I had to write a schedule which puts in paid activities like... Tues am, swimming, but also a whole list of free/cheap activities to give the nanny ideas, also suggesting a few to do each week works well.
The only thing I've had a problem with (now I found a lovely nanny) is tidying at the end if the day, which is due to timekeeping rather than laziness, so am in the process of working that one out. Will probably write a detailed end of day schedule and trial it, make adjustments & make it a habit when I know it works. It's that dinner, bath, bed with tired children & nanny that makes it so hard I think.