Here's OP's pics:
If I read that right, the non-working solenoid worked when you triggered it from the working one's Uno pin. That was a good test.
You could have gone one step further, and changed the code and wiring to use say pin 11 instead of 13. If that works, you're sorted.
(Except for not knowing why 13's not working and I'm wondering if that's to do with the built-in led and not having high enough voltage at the pin itself, to be seen as high by the 298. But try 11 or something first, and if that works worry about 13 later.)
edit: Just check voltage on pin 13 of my genuine Arduin Uno and it's just under 5V so not a problem with mine, anyway.