At what point during a successful run does it officially end?

I am interested to know if I can apply the hosted agenda counter on a scored House of Knives after the runner successfully accesses an advanced Project Junebug or similar card. Does the run end when the runner successfully pass the last piece of ice, or when they make the decision to access a card in a server, or after all accessed cards are resolved?

Consult the timing chart in the FAQ. You can see that a successful run ends at step 5 of the timing structure. You can also see that the last opportunity to use Paid Abilities is step 4.3 - after the runner declares to access (declining their last chance to jack out), but before the access actually occurs. House of Knives is a paid ability.