The Upper Klamath is always in, but it is a logistical nightmare and not an ideal kayaking run. It's filled with jagged pumice rock and not very friendly... but people do it.
The Illinois is a crap shoot but worth every mile if its running.
The upper sections of the Rogue are nice. Takilma Gorge, River bridge, etc. all are on the way to Crater Lake. If you're in Oregon, Crater Lake is a mandatory stop.
Then you're looking at the Umpqua and at that point you should just keep pushing North to hit the White Salmon.
I don't envy you, yours is a hard decision.