I have a 9400, so your cradle is going to be quite different: I have seen a number of occassions where the red LED does not light up but the unit still charges. Jiggle the little Pronto guy in the cradle and the contact points shift between the cradle and the TSU: Voila the red light comes on.

But it sounds like you have a dead battery..... Lithium Ion batteries are real workhorses; however they do tend to bite the dust hard after about 2 years of good heavy use.

Philips does sell replacement batteries for the TSU's, you should be able to get one through your local distributor or on-line (not through Philips). I'm not sure about the 9600, but the 9400 has a neat little screw on the backplate that if memory serves allows you to access the battery compartment.

PS: I agree with your assumption: If the white LED is coming on then the base is getting power. Thus the TSU should charge in cradle..... Unless there's no Lithium left in the Ion ;-)!