Hello & welcome,
From the symptoms you have mentioned here it is clear that all you got is the abscess
related to your broken tooth.
This is generally related to the infection in the open pulp tissue of the broken tooth which makes the surrounding tissue swell due to pus formation.
If you feel that after pressing on the swelling, your nose runs then it may be related to or infection must have spread up to maxillary sinus
which should be treated immediately.
First, visit your dentist
without waiting long & x-ray must be taken for the same tooth to check the exact damage by the abscess present.
If there is no possibility of the tooth being saved by Root canal treatment
followed by post & core with crown, extraction of the same tooth might be necessary.
Antibiotics & Anti-inflammatory drugs carry vital role here along with incision & drainage
which will be performed by your dentist.
Avoid hot fomentation & intake of hot drinks as it may add to the situation.