I don't know how our knowledge of biology compares, but I'm starting my second year in October so you probably know more on the subject than me. I was on the common year one for all subjects within the school of biomedical and health sciences, so I studied modules pertaining to physiology, biochemistry, histology, pharmacology, etc.
The main problem with any method of targeting cancer cells is that it's very difficult to differentiate them from healthy cells as any number of mutations could be causing the uncontrolled mitosis to occur, and as you said they can continue to mutate further at an alarming rate. But it appears you've studied the topic in more detail than I have, so you can ponder that one
Another thing I wonder is how the nanobots would go about destroying the cells? The body's cytotoxic lymphocytes (NK Cells and some T-Cells) trigger apoptosis by using perforin to perforate the plasma membrane and releasing proteases into the cell; the products of decomposition by these proteases act as chemical signals to macrophages to destroy the dead cell.
But without a large enough quantity of the correct chemicals (probably hundreds of times the volume of the actual nanobot, depending on its size), how could it hope to be able to trigger apoptosis in affected cells? At best, we might be able to synthesise basic equivalents of the body's cytotoxic cells, which would trigger a whole range of other problems, one of which being rejection. Begs the question: why not just work on modifying real NK cells so that they're not rejected?