I’m a Londoner and I live close to where you’re staying, hi!
Unfortunately there is no easy or fast way to get to Kensington palace, but you have a couple of options, depending on your preference.
My preferred option would be the bus: the 148 from Elephant and Castle. You can walk to Elephant and Castle or get a bus down the New Kent Road; buses 1, 188, 53, 453, 63, 363 and 172 will get you there in 5 minutes.
You will only be charged once for as many bus rides as you like within an hour, so you might as well jump on a bus to Elephant.
At elephant you need to find stop Z, the London college of communication (if you get the 53 or 453 to elephant, they also stop at that stop, so if you’re worried about getting lost stick with those 2!). From there the 148 will take you to victoria gate, a pleasant 20 minute walk through Kensington Gardens to the palace. On the way you go over Westminster bridge so you’ll have a nice view of Big Ben!
The alternative is the Bakerloo line from elephant to Paddington, it’s a slightly longer walk from Paddington to Kensington palace though. And no nice views!
Hope that helps. Feel free to message me for any local recommendations for the area too!