The battery probably isn't a part of the power-filtering system. If you aren't sure if you'll be able to use the laptop without it, just disassemble the laptop, remove the battery and power on the computer. If it works, then there's nothing to be worried about. If it doesn't, firstly check if you have connected everything properly; If everything seems OK, then insert the battery again and check if it changes anything - if it will work then but not without the battery, just leave it inside.
Before sending the battery to the recycling center, i would suggest to try to replace the cells - you may make it work like a new one! Firstly, if you don't feel confident in electronics, don't do it - this process will include desoldering existing cells and soldering new ones, which is kinda dangerous if you don't take precautions, and may be hard for a beginner.
Before I even start on how to do it, i must write that i give no warranty that you won't destroy your laptop/battery and/or injure yourself. Be careful.
- A clean desk
- Lots of time, patience, and some soldering skills
- Flat and/or phillips screwdriver
- Soldering iron
First, if your battery is put together using screws, unscrew them. If you don't see any, then carefully insert your flat screwdriver in the slit and pry the battery open. Be patient with this step, try avoid damaging the plastic.
Second step is to determine what cells your battery uses. They should have a numer like 18650 on them (but they probably won't be random 18650s - 18650s are LiIon, not NiCd). Search for that number on a site like Amazon or Ebay; If you'll get any results, buy an exact number of cells that you already have in your old battery; New cells can be rated for a little bit higher or lower voltage (~±1V is still OK, but anything higher/lower than that won't be good for your laptop), and can have a higher or lower Wh/Ah rating - this one will only change how much uptime will your laptop have on the battery.
When the cells will arrive, desolder the old ones from battery circuitry (from the Printed Circuit Board, or PCB), and solder the new ones exactly like the old ones were soldered. Don't forget to check the polarity of all batteries you solder - you should connect + from one cell to - from another cell, or if some of the cells are connected in parallel way in the original, then + from first cell to + from the second cell, and - from the first cell to - from the second cell (this way two or more cells will work as 1 bigger one). Don't forget to double-check if all cells are connected correctly before soldering two leads to the PCB.