With that kind of budget the world is your oyster for an extended stay, lol. Seriously dude, you'll have choices.
Hopefully the water issues won't be *that bad* on the west end below Summerset Lane. Us folks deep west are getting the worst of it.
You are also smart to start looking now - January and February are really busy months and lots of folks come down around then for several month stays.
Off the top of my head you might want to check Seastar if they are not booked up already. They have some nice rooms with kitchens, good internet and water pumps and tanks .
Catcha Falling Gardens has one "regular" room with a kitchenette and an apartment with a more full kitchen to rent - and its not unusual for them to rent either for a month.
Citronella might be another choice or Villa Sur Mer - they may give you a break on their nightly rate for a month's stay but it might go over your budget...still, both are lovely and worth checking out.
Hit up Debbie at Moonlight Villas too. I think she might have an apartment with a kitchen that she rents - not 100% but check her out.
So that should get you started, at least West End way. On the beach you might want to look into Negril Beach Club (am I getting that name right?) - they rent condos...or Point Village, the owners there rent their condos as well.
Happy trails!