Flight Instructor with student reported a near miss with another aircraft in the traffic pattern. The instructor took evasive action to avoid a collision and was later issued a go-around by ATC for sequencing.

Narrative

We were in the right downwind for Runway XX at 2500 ft. msl. Instructed to begin descent to TPA (1600ft) by Tower. After beginning descent; I noted using my ADSB IN that there was traffic a few hundred feet below us coming out of a LEFT 360 turn out of the RIGHT downwind XX. No traffic advisories were given to us in the descent to be looking for traffic nor were any heard given to the other aircraft. After spotting the other aircraft. I decided to take controls from my student and initiate a climb to remain clear of the other aircraft. After initiating a climb; I notified Tower of the conflict. Tower apologized and told me to; 'maintain 2500 and continue RIGHT Downwind they will call our BASE.' Moments later he re-cleared me to descent and to turn base behind a Piper Archer/Cherokee. After following the instructions; Tower issued S turns to the Piper I was following. Tower then requested I do it as well and we complied. Due to us being in different aircraft and having a faster Vref speed; we were unable to maintain distance and I had to alert Tower that we didn't have enough separation for us to be able to continue. Tower issued me a Go-around right side and join the right downwind again further apologizing. The rest of the flight was uneventful.