Run-on, comma splice or just too many clauses?

I'm grading my student's paper and this is his sentence:

Without wasting any time, Lady Osbourne took the boys with her after bribing the head lady and they set off in a fleet of luxurious cars back to her home which was located amid the most elite neighbourhood full of wealthy business tycoons.

Is his biggest mistake too many clauses in one sentence? Or is it a run-on or a comma-splice?

My instinct says that I should tell him to simplify his sentence, and separate the clauses. Should I follow my instinct?

It needs a comma after the "head lady" and another after "her home."

But it is certainly unwieldy even after those commas are added.