|A slight diversion: Why have you removed the battery; is it just because it doesn't hold its charge completely or has it utterly failed (internal short circuit perhaps).|
I find the latter to be very rare - in fact have never experienced this myself.
Just my opinion, and I'm happy to be corrected, but I have several old laptops at home and within my circle of family/friends all running on mains but with tired, old batteries of varying efficiency still installed.
Some hold their charge for only a couple of minutes, but usually, this still gives a level of protection against mains power cuts and glitches (similar to an interruptible power supply for desktop PCs) and usually with enough reserve to shut down the laptop safely (hibernate), or at least give a visual 'power' warning.
See how you get on with the previously- provided suggestions, but unless your battery has suffered some kind of catastrophic failure, I'd be tempted to leave it in situ.