The young winger found the net twice in the first half to help the Eagles recover from their setback at Fulham on Saturday afternoon.
It took just two minutes for Kaikai to test the hosts' backline, but he was denied on the first couple of occasions by a mixture of solid defending and smart goalkeeping.
Bromley saw a couple of half-chances of their own pass them by as the match wore on, and that proved costly when Kaikai swept the ball home following a Jonny Williams knockdown inside the box 20 minutes in.
Robert Turgott squandered another decent opening for the National League side prior to the interval which, once again, would haunt the home side as Kaikai sent a free kick in off the post to double his tally.
The chances dried up in the second half in front of the 3,646 watching spectators, with very little goalmouth action at either end, but Bromley did give their supporters something to cheer about at least when Daniel Ajakaiye picked his spot late on.
Palace face Valencia on home soil next weekend in their final run-out of the summer, before playing host to West Bromwich Albion in their opening Premier League fixture.