It appears you're not the only one with that problem; it seems to be common enough that there's even a YouTube video on the whole procedure of disassembling the unit and replacing the switch.
[link|https://www.youtube.com/watch?v=3lIzCLCqoZU|Philips Sonicare Elite Toothbrush Repair: How to Replace On/Off switch - YouTube|new_window=true]
You'll need to be comfortable with a little soldering on a relatively small part mounted on the circuit board in order to accomplish this repair, though.
You might check around on some other videos for the part; although the comments state that you can buy the part on the person's eBay store, that store is no longer in operation. If you can't locate the specific part, it can still be found fairly easily just by measuring the length, width and height of the button in millimeters and putting that into a search engine, something like this.
[quote]6mm x 6mm x 3mm SMD push button switch[/quote]
That's just an example since I don't know the actual dimensions of your particular switch. Pick one that looks as similar as possible to your existing switch in terms of the height of the button and the placement and shape of the circuit board contacts and you should be good.
Be sure and let us know what you find, that way someone else in the future who has the same problem can benefit from your efforts as well. Even better if you get ambitious and want to write your own guide here on iFixit!
[link|https://www.ifixit.com/Guide/new|Create a new Guide - iFixit|new_window=true]