Oddly enough, I've been having the same problem with my Toshiba laptop. It's a Satellite 1995 S803(maybe 805, can't remember off the top of my head). Basically it won't automatically stay on and charge when the power supply is plugged in. I can have it in for a little while, but if it gets jolted it cuts out and you can't reconnect to get it charging again. Sometimes it just cuts off the charging supply whenever it feels like it. I either have to use it until it drains out and then recharge it, or turn the damn thing off and pray that I connect the charger in the perfect way and turn it back on to have it charge while I use it. I've pretty much given up on the damn thing and am just going to keep doing it that way until it finally gives out. I have a desktop PC as well with backups of everything.
How long have you had the laptop? I would just say go and get another cheap one, especially if all you are going to do is use it for notes and internet browsing and just use the other one occasionally.