Logitech 350 keyboard

According to Logitech, that keyboard has two "AA"batteries. Have you tried changing batteries?

Logitech has posted a rather nice help page for that specific model here:

https://support.logi.com/hc/en-us/articles/360024328973

Might find something helpful there.

Unplug the exrternal keyboard, use the internal MBP keyboard an

Try resetting NVRAM/PRAM http://support.apple.com/kb/ht1379

replug in your external and test.