chuck,
I know it's been a couple months, so maybe you've figured this out, but I'm going to share this, as I'm seeing that this apparently seems to be a defect on many Dell XPS M1210 laptops.
my m1210 went out in November. I dismantled it, and upon inspecting the motherboard, I found that component P121 on the charging circuit had fried. It literally sparked on the motherboard when a power supply was connected.
The result is that even though BOTH your battery AND power supply are fine (I'll bet if you swapped them with your other friend with a 1210 she wouldn't have any issues), the charge circuit is broken and power dose not get from your charger into your computer.
The bad news is that your motherboard is going to need to be replaced. The good news is that there's a temporary workaround that will allow you to use your laptop, charge your battery, and limp along until you save up the $300+ bucks to buy a new motherboard.
Plug the laptop into a monitor.
Yep, you heard me. Whomever designed the motherboard for this laptop did something interesting. They routed the very same circuit that failed on your motherboard through the monitor connection. The result is that as soon as you plug your laptop into a monitor, it will bypass the failed circuit and spring back to life, allowing you to use and charge your machine. The downside is that because you need a monitor to charge up, you can only use your laptop away from a monitor for a couple hours.
I'm trying to hack together a dongle you can plug into your monitor output to allow you to charge without dragging a monitor with you everywhere you go. I'll post instructions on building it on my site as soon as I figure out which pins do the trick. Really though, it' probably just easier to buy a new motherboard. If you could see the spark I saw, you wouldn't want that happening every time you plugged in the thing.