You could try taking the remote apart and cleaning the contacts etc, assuming you tried multiple sets of batteries and rebooting your 3600, etc.
However, you likely need a new remote: Either the $20 Voice Remote (RCAL7R) or $30 Voice Remote Pro (RCS01R) can be used, and can be obtained from Roku.com/Walmart/Best Buy/Amazon/etc.
https://www.roku.com/products/accessories/streaming-stick
However, considering the age of your model 3600 and the fact that you can get a new $30 Express 4K+ or $40 Streaming Stick+, I recommend just getting a new Roku:
https://www.roku.com/products/players