I wonder if this might be a solution: The executive branch's power, in general, has grown beyond what was originally envisioned. So, to mitigate this, remove the involvement of that branch entirely. Give it to the legislature thus: they must appoint a panel consisting of judges who's responsibility is to appoint new Judges. Each side would appoint an equal number to the panel, with a final single appointee appointed by agreement of both parties in the Congress. If no agreement can be reached on this appointee by a specified date, the "partisan" panelists would take over that responsibility. If they cannot agree, then they would be barred from being nominated again to the panel permanently, and the process would begin again.
Presumably, over time, the quality of judges would improve as far as non-partisanship. Thus, the panel itself would improve, and the process would deadlock less often.
I envision the panel holding the responsibility for appointing Supremes as well. I believe this system would restore credibility to the judicial branch over time, which would be an awesome thing.