What the failure actually did
Your cycle lives in the filter, on the surfaces of its media, and those bacteria need two things the failure just cut off: flowing, oxygenated water and a steady trickle of waste to feed on. In a stopped filter the colony starts declining within hours, faster if the media dries out or sits in stagnant water that has used up its oxygen. How much of the colony survives depends mostly on how long the filter was off and whether the media stayed wet, which is why the first moves are about preservation, not replacement.
Assume the tank is now under-filtered even if the fish look fine. Waste production has not paused just because filtration did.
While the filter is down or being replaced
Move the media into a bucket of water taken from the tank and keep it submerged; if you must clean it, swish it in that bucket water only, since chlorinated tap water would finish what the failure started. Aerate the tank itself with an airstone if you have one, because the fish have lost the filter's surface movement as well as its cleaning.
If the unit is repairable, repair it around the saved media. If it must be replaced, run the old media inside or alongside the new filter rather than starting the new box empty; the surviving bacteria will seed the fresh surfaces from there.
The careful fortnight after flow returns
Feed sparingly for the first week or two, well under normal rations, because every meal is ammonia arriving at a colony that is rebuilding its numbers. Test ammonia and nitrite daily, at the same time, and write the numbers down rather than reacting to colours from memory. Small rises get partial water changes with conditioned water; the goal is to keep the fish safe while giving the bacteria the modest, steady food supply that regrows them.
Resist deep-cleaning anything during this window. Gravel vacuuming, filter rinses and media swaps all remove bacteria the tank cannot spare right now, so the tank may look slightly less polished while it becomes safe again.
Borrowing strength and knowing you are done
The fastest legitimate accelerator is mature media from another healthy tank, tucked into the recovering filter, which is effectively a transplant of working bacteria. Bottled starter cultures can add a little; borrowed media adds a colony. Either way, testing decides when you are finished, not hope: the cycle is re-established when ammonia and nitrite both hold at zero across several consecutive days on normal feeding, not on the starvation rations of the first week.
Ease feeding back up in stages while the tests stay clean, and if a reading spikes at any stage, drop back a step rather than pushing through.
Charting each day's pair of readings in the tank's Pet Passport on KinsLe shows the stall and the recovery as one clear story, and tells you the exact day normal feeding became safe again.