The Supreme Court has an issue with recusals. They don't handle them well. There is no one above them to say "Hey, you can't hear this case." Recently a lot of people are complaining about their inability to conduct recusals fairly ((Source)
Part of the problem is the precariously balanced right/left nature of the court. A judge can't recuse him/herself without dramatically shifting the balance.
The current US court system works as follows: 94 district courts, plus the US Bankruptcy Court, the US court of International Trade and the US Court of Federal Claims (suing the US government). Above them are the 12 regional courts of appeals plus the "Court of Appeals" which covers the Bankruptcy, International Trade and Federal Claims court.
Then there are the national courts not part of the Appeals court hierarchy. These include Military Courts (trial and appellate), the Court of Veterans Appeals, the US Tax Court, and certain administrative agencies and boards. These typically do not get appealed to the Supreme Court, but theoretically the Supreme Court could claim jurisdiction over a case they try. In addition, there are the various state court systems.
Above everything is the Supreme Court of the United States (SCOTUS) They are the top.
This is what should be done;
First, we need some one to appeal a SCOTUS Judge's refusal to recuse. I suggest a board of judges randomly chosen from the Court of Veterans Appeals, the US Tax Court, and the state supreme courts (Note, in New York, the 'Supreme Courts' are actually the lowest state court and the highest level court is the "Court of Appeals" - in this and similar cases I mean the highest state court, not the one with the word "supreme" in it.) Recusal requests should not common, so we could just take all their names and select five of them at random to hear the request.
Second, we can not allow the balance to be overturned. As such, when a Supreme Court Judge chooses to recuses himself of his own free will, then he can select any judge from the 13 Federal Courts of Appeals (that was not previously involved in the case) to stand in his place. Give him the total control over this process, assuming he willing recused himself. If however, he has refused to recuse himself, then the judge will be selected randomly (from the Federal Appeals Court judges not previously involved in the case).
This is the key thing. By doing it this way, it will encourage the judges to recuse themselves when appropriate, as they get to pick their replacement.
In addition, it will give the Appeals Court Judges some "Supreme" experience, which may make selecting a replacement when a SCOTUS judge retires just a little bit easier.