You can only paddle the Don during the annual "Paddle the Don Festival" (which I've wanted to go to for sooo long - apparently spots fill up super-fast). Normally, the river is too shallow in parts and you would need to portage/drag through. But for the festival, they release a gate of water onto the river in the morning so paddlers can enjoy deeper water for basically their trip down and their trip down only.
Likely some parts are deep enough any time of year, but you wouldn't get much of a day's paddle out of it.
Sort of irrelevant but I paddled the Grand a while back from Cambridge to Paris and it was really pleasant. If you're just looking for a close-ish river to paddle.
And a friend of mine worked at an Outdoor Ed centre owned by the Toronto District School Board on one of the islands a while back. Maybe that's where the Scouts camp? If you look at islands other than Centre Island, you may have more luck with camping.