I tentatively suggest that the emphasis of the “haud” as opposed to a milder negative like “non” is what’s motivating the forcefulness of the translation.
Colloquially, it might come across as something like “No way am I changing the done [deal].” “I certainly [would] not change [what’s] done.”
I agree that translating that with “repent” seems a bit free, but it has a pretty firm sound to it and maybe that was the idea. I’ve also seen “I don’t regret it”.
While there are indeed tons of Google hits for the translation “Nothing happens by being mute”, I don’t know of any actual use of the phrase in Latin where that meaning applies. (In fact, the only occurrence of “haud muto factum” that I know of at all is a passage in Terence. Maybe it’s a more common idiom than I realize or maybe it’s just a one-off literary quotation.)
ETA: The “nothing happens by being mute” translation references all seem to trace back to some journalist’s description of the aristocratic family motto of the late UK fashion designer Isabella Blow (d. 2007). Did somebody at that time just naively mistranslate a late-medieval motto taken from Terence and it’s been wandering the interwebs ever since?