I don't know what sights the tour covers or what sort of things you're interested in, but these are some of the big-deal sights covered by the pass that happen to be rather expensive if you have to pay out of pocket:
Tower of London
Westminster Abbey
St. Paul's Cathedral
Hampton Court Palace (travel time required for this one)
Kew Gardens (travel time required here, too)
A couple of sights in Greenwich could be combined, so travel time would be shared.
There are many other options, of course.
I don't know exactly how the London Pass works; there might be a reason not to use it for one or more of the sights I've listed--such as if there's a requirement to book an entry time that's challenging to accomplish with the Pass.