I knocked over a small glass of water on my laptop last night, most of the water skirted off. I dried the computer by taking the keyboard out, opening the case from one side and using a hairdrier on low.

Here are the problems: Keyboard is typing by itself, types double letters/number and some keys dont work at all. I tried reseating the cable, but this did not help the problem.

Power - The main power button does not work. I can get the pc to power on by taking the battery out, unplugging the power cord from the back, then plugging it back in. If the battery is in or the pc has been shut down without the battery installed and with the power cord in, the pushing the power button does not power on the pc.

Please let me know how I can approach fixing these problems.

and the laptop is HP? anyway, we need the model name.
what would you do if you spilled some liquid on keyboard?
1. shut off the unit immediately!(that means, plug out AC cord and battery as faster as you can).
2. turn it upside down(if it's possible open it and lay it on the table upside down, thus the liquid won't spill LCD module).
3. after some time, start to disassemble it fully and clean inside the laptop.

in your case, the liquid is already damaged power chains inside the laptop, and you'll have to fully disassemble it and clean!