thcri,
I have what sounds like an identical setup to yours, A Dell system about 1.5 years old, and I think a logitech wireless optical mouse and keyboard. It was supplied by Dell, but I'm 95% sure it's a logitech (I'll check when I get home). Anyway, I get about 6 months out of a set of batteries. I really like the keyboard and the mouse, and the fact that it can be dropped in a desk drawer out of the way of little fingers and sharp little eyes.
One thing to check, and I must confess I don't know much about this, is the light on the base station. When my mouse went dead the last time, I was confused and thought my computer had locked up. But then I saw the red light on the base station and realized my batteries were dead. Popped some new ones in and all was well.
I think your mouse is defective. Did you buy an extended Dell warranty?
EDIT: I just realized that I've never replaced any batteries in the keyboard. I assume yours is still working?