The AC-Supply ***SHOULD*** be capable of running the computer even with the battery removed... I know I have run some older computers that way.
And the battery should not make any difference to the Wi-Fi, but the key word is SHOULD, it might, epically if the mode of failure was causing RF-Interference.
In any case, take it to a Batteries PLUS store,,,,, Odd are they can fix the battery.
Just got me a new Toshiba C55D at Staples for 250.. That's not far from the cost of a new Laptop battery these days (Around 75-150) heck it's only 100 less than the cost of the hard drive that is in it (500 Gig Laptop drive). Nice box.
Oh, that price... Deal of the week, GOOD TODAY ONLY. goes up to I think 400 tomorrow.