I have found solution for my problem. My problem was:
Keyboard was not displaying both in BIOS and Windows. I've tried installing keyboard fix from Dell, tried Power management from Dell. Nothing helped.
I have tried to charge Travel keyboard both through built-in USB and tablet -- nothing helped. Tried different cables, tried upgrading/downgrading BIOS - zero result.
I've tried to reincarnate battery several days by trying different scenarios...
But suddenly on 3rd day it started working, I absolutely don't know why. I left it plugged into charger (keyboard connected to tablet) and probably that helped, but I swear I tried it before too, but this time keyboard battery somehow started working.
But strangiest part that LED indicator on keyboard still flashes only for 30 seconds and turns off, while making bzzz sound. But with tablet connected it charges.
I've already been thinking of buying new battery, honestly. But fortunately it started working again.