You can try St. Augustine. I'm not 100% sure they'll let you if they don't know you, but you can ask. You would need to come in through the office at the side.
Alternatively, there will be Mass tomorrow (Friday, and generally every Friday) at 12:15 and then Adoration immediately afterwards until Benediction at 3. It's a quiet time that anyone is welcome to attend as the church is open (you can enter by the main doors), and you won't be disturbed. You might find the quietness that you're seeking, we believe literally in the presence of Christ.
Here's an outline of what it is:
https://en.wikipedia.org/wiki/Eucharistic_adoration
Personally, it's my favourite time of the whole week when one can sit in contemplative silence.