You're asking for a complete refund for an all-inclusive multi-day stay at the resort, plus $2,000 on top of that? You would have to prove substantial damage for that to be awarded to you in court. So what could you be owed?
Bookit is a reservation service (travel agent) and almost certainly didn't retain your money - the vast majority of it went to Princess Hotels. Bookit would likely offer you a small courtesy refund, but if you believe you're owed a large refund, it would be Princess Hotels that owed it to you. Bookit might advocate on your behalf if they felt strongly that the hotel was seriously sub-par, however, and act as the conduit for any financial compensation.
Here's what you will need to provide:
1) Details of each contact you made with Bookit enumerating serious problems with the hotel and asking about options to be moved to a different hotel.
2) Dated pictures of the fleas and dated pictures of the bites on your bodies.
3) Medical records of all required treatment and doctor visits, and/or pharmacy receipts of medications and salves bought to relieve the bites.
I'm not sure what to say about the laundry situation. Hotels usually charge for laundry. If the issue was that you were concerned there might be fleas in them, then they would likely have just put them through the dryer or with a dry cleaner pad added to the dryer.
I'm so sorry about your husband's medical condition, but although there certainly shouldn't be a cost to washing the full set of sheets, if there was any leaking on to the mattress, the hotel might have in fact been charging for the necessary treatment to clean the mattress. I have a friend with similar issues and she is careful to wear adult diapers in hotels to avoid exactly that situation.
On the face of it, I think a settlement from Bookit and (mostly) the resort itself should be $200-$400, not $4,000-- unless you do have evidence of real harm plus details of attempts to be moved due to unsanitary conditions, and the refusal of both/either parties to accommodate such a move, in which case, you should attach them here.
If you're not happy with whatever they do offer you, you might try calling your state's Attorney General to ask if they have any suggestions about making a complaint regarding an overseas hotel.