You’re not far from the Leeds-liverpool canal at that hotel, not ran there myself but in Leeds city centre it’s the best place to run IMO. It looks like there’s a decent path next to the Mersey too.
That’s what I did in central London when I was there overnight last year; running along the Embankment early evening though…that was a mistake! My early run the next day though was great; running down The Mall and it was pretty quiet; soggy tourist map in hand.
The website for the university sports centre is indeed unhelpful. But the swimming pool is open to the public (old proof search instagram BrDtJ7nFX5t sorry I can’t add link)
Best running running options from there - probably head to and around Princes Park and/or Sefton Park.