I finally decided to purchase a replacement battery on eBay, I just installed it on my K750 keyboard (again, damaging the battery compartment even further...), and the keyboard seems to be working fine again! Solar App tells me that the charge is already at 100%, so I hope it'll keep that way for months to come (like the original battery did).
After a few weeks using a different keyboard, all I can say is that I'M SO GLAD I HAVE MY K750 UP AND RUNNING AGAIN!! <3
I can't find this particular battery anywhere for now, but since my keyboard died and doesn't turn on no matter what (sometimes the led blinks red when I turn it on, sometimes it doesn't lit at all, but it doesn't seem to be charging), I tried to take the battery tray out anyway, following instructions here and elsewhere. I found it very hard to remove and I damaged the tray and the keyboard while doing it. After re-inserting the battery, the keyboard is still dead :( I hope when I get a new battery it'll fix things, because I've grown addicted to this keyboard, it's the best one I've ever tried because it's soooo comfortable to type on it. Also, Logitech sucks so much for this battery issue, I hope they go bankrupt one day >:(
Anyway, here's another take (more detailed) at the whole removing the battery tray process: