The 31-year-old midfielder was the subject of intense speculation throughout the summer which continued up until deadline day.
Reports suggested that Ian Holloway - his former boss at Blackpool - was keen on a reunion at Crystal Palace.
No deal was agreed, though, and Vaughan is now looking to cement a starting berth in Paolo Di Canio's new-look side at the Stadium of Light.
He told the Sunderland Echo: "Obviously, there's been a lot of speculation over my future and stuff during the summer. But I just want to be playing football.
"I'm pleased to be back in the team at Sunderland, but hopefully I can improve on the performances from the last couple of games."
Vaughan has started the Black Cats' last two fixtures, but has been unable to prevent them from stumbling out of the Premier League blocks and collecting just one point from their opening three games.
He added: "We've got to try to pick ourselves up and get some points on the board quickly.
"The international break has come at a bad time, at the start of the season. But it gives us a break mentally and hopefully we can come back fresh for the Arsenal game.
"We need to stop conceding poor goals from set pieces, try to get on top of games early on and just give ourselves a chance really.
"We seem to be switching off at the crucial times. We should be doing better than that."