Private Pilot flying C-172 aircraft reported brake failure on landing rollout.

Narrative

I departed ZZZ1 at approximately XA20 local; after refueling and a thorough pre-flight inspection which identified no anomalies. The flight was conducted under Part 91 VFR rules. I am a Private ASEL pilot with approximately 450 Hours. My sole passenger was a non-pilot.At approximately XC00 local; I arrived in the airspace of ZZZ; and contacted Tower requesting landing; with information Yankee. I was instructed to join a right downwind Runway XX; which I complied with. Upon reporting right midfield Runway XX; Tower cleared me to land Runway XX #2 following a Beech on a 10 mile final and to extend my downwind. I complied.Upon passing abeam the Beech; I noticed a second plane closely behind the beech; and verified the tail # on Foreflight as another aircraft Tower was controlling. The other aircraft was an Archer. The Archer was instructed to make a right 360; which he did for spacing; and he was cleared to land #3 Runway XX. I had visual separation with both aircraft and turned right base and then final.Upon touchdown; I was instructed to exit Taxiway X (on the right); following traffic Short Final - this would normally be a non-issue; with the application of medium brakes. As I applied the brakes; the right brake pedal/rudder went to the floor and I only had braking action on the left side. It took all of my concentration to keep the aircraft from veering to the left and off the runway; as I figured out how to recenter the aircraft on the runway and attempt to slow down. Upon passing Taxiway X; Tower issued me exit Taxiway Y (also on the right); no delay; traffic behind on short final. I assumed that I would be unsuccessful at making a high-speed right turn off the runway; so I slowed as much as possible; swung to my left; and then applied full right rudder plus light left brake. As my aircraft was turning from the runway onto Taxiway Y; Tower issued a go-around to the Archer because 'aircraft (me) still on the runway'.I successfully stopped the aircraft after the hold-short line on Taxiway Y; and contacted ground for taxi clearance to the 'West Tie Down'. I was issued Taxi Z; A Cross Runway XY. It was difficult to make right turns without a right brake. I was still a little surprised by the incident; however I thought that I should inform Ground Control that I had a mechanical situation on board; and that they should check the runway for FOD/parts/brake fluid. I was told to hold my position on Taxiway A between Taxiways Z & B. Airport Operations & Rescue came to me and followed me to the West Tie Down. A liquid; similar in appearance to brake fluid; was observed on the inside of the R main tire and throughout the inside and outside of the wheel pant. Fortunately; no parts or significant amounts of fluid were on the runway or taxiways. What should I have done differently? I should have informed Tower that I was unable to make Taxiway X; and then to inform that I had a failure while I was still on the runway; as the aircraft very close in trail was already cleared to land. In my defense; I was aviating before communicating. I should have also not proceeded with my taxi until the aircraft/ground was checked to ensure that I was not contaminating the runway or taxiways.