With the chef's authorization code, we've almost got everything we need to reach the hand.
We need something sturdy, and something else to scrape the hand off the ceiling.
Most cooking utensils are adept at scraping - maybe our chef-bot friend could spare one after he's done cooking?
Well, it doesn't hurt to ask. The boys are probably getting hungry anyway.
Ask him to prepare scrambled eggs, then grab the spatula that he throws.
As anyone human-or-otherwise that's familiar with technology knows, sometimes you need to give it a good WHACK.
Search the technology-focused rooms.
Return to the server room, and take the mop stick from the left side - see these hints if there's a guard in your way.
By combining the spatula and the mop stick, you've got the perfect tool to reach the hand.
Use it to free your imprisoned friend!