Too late now most likely, but it's useful to have one Tom Jones song *locked in* and ready to go in your back pocket for random karaoke. The hammier and cheesier you make the performance the better I think Delilah is *perfect* (it's so over-the-top, and it's only 3:30), but It's Not Unusual can work as well. The secret to karaoke is that getting into it and acting cheesy stands out *less* than the person who's totally scared and mumbles their way through stuff. Lean into it. Or, when in doubt, Bon Jovi's "Dead or Alive" can't miss.
The Greatest Karaoke Move Ever(tm) is done by my friend who lives in LA; he's the British-est Englishman to ever British a British. But he actually found two common korean ballad that all of the karaoke bars *always* have, and learned them perfectly (because apparently in LA basically all karaoke bars are Korean). So the first time he goes out for karaoke with a group of people, he acts all shy and demure until they eventually force him to go up and do a song, and he always busts out that song. Jaws, commence hitting floor. Then if he goes out with them again, he's got the second song ready to go.