I’m not even a real lawyer, let alone a Constitutional scholar, but here’s my take:
First of all, if the corrupt Justices have enough allies and support then this becomes a constitutional crisis and/or civil war, at which point the answer is laws, schmaws, who’s got the guns? So we’re assuming that the bribers blew their entire budget on the Justices, and pretty much everybody else is now trying to reverse this clear perversion of justice.
I see a couple possibilities: One is that the losing party asks for a rehearing, requesting all the bribed Justices to recuse themselves from the rehearing. This requires some measure of shame and remorse by the bribees, but is the quickest solution.
In any event, I’m pretty sure it’s a federal crime for a sitting federal judge to accept a bribe, so whichever US Attorney has Washington D.C. in his jurisdiction could send the bribees off to Sing-Sing.
At which point, there will have to be some kind of decision as to whether a federal judge can be removed for being bribed and/or imprisoned. In the absence of any specific existing statute or procedure, it would probably end up being the SCOTUS that decides, which brings up the point of to whether the bribees would participate in that decision. Likely, the unbribed Justices would just say “We’re the only ones who matter, and we vote to throw these corrupt bastards off the court; just ignore whatever they say about their votes.” And everyone would say “Well, maybe some law professor might say that’s not a 1,000% kosher ruling, but we’re all going to go along with it.” and the US Marshalls would drag the bribees out of the building and into a cell.
And then finally, Congress does have the power to make laws about how the federal judiciary works, so they can (and quite possibly have already) pass a law saying, essentially, “Federal Judges are no longer considered to be “in good behavior” if any of these criteria are met (one of which is being convicted of a felony involving their office), and will be removed from office following this procedure…” The procedure would probably need to involve decisions by other judges, rather than say the President, but would probably include a clause disallowing any judge from ruling on his/her own removal, so we’d get to the point above, where the remaining Justices kick the bribees off the court.