For further clarification, my ideal system would be that should the house choose to impeach a supreme Court Justice and then the supporting of their impeachment would be brought to popular vote by the American people.
Sure that's a little clumsy, but it allows for not just political motivation to rule the Day when it comes to correcting our supreme Court and checking their power.
Quick edit, I would probably also make this a required vote at every congress. At least once a year the house has to vote whether or not to keep the current justices in power or to turn their Fates over to the American people.