You can see the iconic sites all on your own. Use your tap credit or debit cards ( 1 for each of you) and use the London Underground. So easy. Take a Circle line train from QUEENSWAY to Westminster station.
Exit and you’ll be able to see, a 1/2 block away, the iconic Elizabeth’s Tower with the Big Ben bell. From Westminster cross the road at light ( with countless others ) and walk past Parliament. Westminster Abbey will be across the road. This area is known as Parliament Square. Meander on your own.
Personally I’d look at the London Walks website and see if you can time an arrival in this area so that you join a tour of the Abbey. www.walks.com
Lastly enjoy a lunch and/or a pint at the Two Chairman pub. Relaxed, walk out the front door, turn right, cross the street and talk the cockpit steps into St James Park. From there take Birdcage Walk up to the Palace.
Been decades since I toured the Palace so won’t offer a yay or nay. Will say a few years back did enjoy a pleasant conversation with a member of Parliament in the Two Chairman pub. Often timing provides pleasant surprises.
Enjoy your time in my favorite city on the planet.