Hmm, I wonder if my battery controller is bad - I remember what happens to it now. It gives me a solid orange charging light, but the battery makes an intermittent low volume, high pitched noise. Checking the battery status in software, it says 80% charged but does not change. What gives?

Update: it has stopped making the high-pitched noise for the first time in the time I've owned it, for no apparent reason. Still have a solid orange light, no charge on battery, and power meter stuck at 80%.

To find out what type of cell it uses you'd have to either measure the old ones or search the web for info. An 18650 is supposed to measure 18mm diameter and 65mm length.

Replacing the cells is simple in theory but can be difficult in practice. First is getting the plastic case open without wrecking it too badly. They are mostly not designed to be disassembled. Then you need to disconnect the old cells. In some cases there are wires that can be desoldered but they probably also have contacts that are welded on which then need to be cut. And you need a way to make the connections to the new cells. Soldering directly to the battery is problematic and potentially hazardous. You can buy cells with contacts already welded on, the only problem then is the extra space taken up by the contact with a wire soldered to it may create a very tight fit inside the case. Aside from that just remember common sense, beware of potential shorts, polarity, and preserve the PCB and any thermal sensors exactly how they were.

There are guides for building a recharging circuit and fitting Li Ion cells into a casing. I haven't bothered to do it myself, as most of the things I have with rechargeable batteries are handheld game systems, and all of my older laptops are dead from physical trauma, like morons stepping on them while I was using them to scan clients' systems. Anyway, not certain how old chargers would read a battery that could have much greater capacity than the original battery, as I haven't looked deeply into it.