I think it's unlikely that your total electricity bill went up significantly because of this. A battery is just an "energy container": the energy it contains came from your wall socket.
If the battery won't hold charge, then the Mac is just using the electricity it needs to run.
If the battery will hold charge, then the Mac is using the electricity it needs to run PLUS refilling the battery. You can of course use the Mac on battery, but then you're recouping the extra you used to fill it.
Let's say your Mac needs 10W to run, and 20W to fill the battery over 2.5 hours. You normally use it for 5 hours a day. With 'no' battery: you use 10W to run it, for 5 hours: 50Wh per day.
With a working battery, you charge up the laptop (while it's off) at 20W for 2.5 hours (50Wh), and then use that charge over 5 hours or so.
Or, you could use it on power for 5 hours, while charging the battery: 30W (running and charging) for 2.5 hours plus 10W (running) for 2.5 hours. That's 100Wh, but now you've got a full battery, which will see you through tomorrow.
TL;DR: Assuming you're spending the same time on your Mac, your energy usage should not change much. But it's worth getting the battery replaced anyway.