I know how exhausting a cat like that can be. We had one for a number of years. Howling at the door, everywhere underfoot, peeing and pooing in any nook and cranny or right in the middle of the rug, wanting in and out a zillion times a day. No fleas tho :) She died of natural causes when i was pregnant so i didn't have to care for her with kids. I can't say we werent relieved, but i still miss her sometimes. when she wasnt a total pain, she was very sweet and loving.
We do still have an older cat (15) who is not neurotic like the other one was but is getting much needier in her old age. Our vet told me that is normal, that they need more attention as they age. i have noticed that even a minute or two of undivided attention helps calm mine down (kinda like kids). It's a pain for me b/c i have a cat allergy and have to wash thoroughly every time i touch her but if it makes her content for a little bit, it's worth it. i explain to my son we need to pet the cat, and often he will "help" (sometimes she doesnt like that as much ;))
Anyway, definitely take care of the fleas- easily treatable, once a month w/ Advantage. i'd have the vet do a checkup for any possible causes or medical treatments that might help
If you know someone who could commit to a permanent good home for a high maint. cat, that is one thing, but there are already many older cats in the shelters and they don't all get homes.
Seems like you need some help with giving the cat attention. Do you have a husband/partner that can help- maybe pet the cat while watching TV or something? Do you have a neighbor kid that could come pet and brush the cat for a half hour for 5 bucks?
My son is 2 1/2 but he will pet our older cat for 30 seconds, i would think by three a kid could do a minute or two a couple times a day- maybe get an M&M for the effort?
Also, it sounds like you just might need a break too- you've got a lot of need to fill- are you getting any "me" time or breaks for you? It's hard to keep giving without getting a chance to relax and recoup for the next round!
One last thing, i vote for the original formula Nature's Miracle as the best product to remove cat pee smell. And a word of advice, DO NOT spray it on a leather couch.... :-O