Some of what's wrong with the Supreme Court is structural. Justices used to drop dead of heart attacks before they aged too far into mental decline. By this point, lots of people have heard about the best solution: replace lifetime tenure with single 18 year terms, with the President getting to select two justices for each election he wins.
What nobody knows, as far as I know, is how to get there from here. How do you work out which Justice gets forced into retirement first to make room for new blood? This could be very hard to work out in a bipartisan manner. (If you have any technical suggestions for how the transition should be managed, please put them in the comments.)
Now that the Democrats have complete power in Congress and the White House, however, they can just go ahead an make this reform on their own. I can't imagine they would, though.
A more subtle defect in the Supreme Court is the lack of adult supervision. We still have the obsolete system of ailing Justices such as 76-year-old Ginsburg (cancer surgery in February) and extremely elderly Justices (Stevens is a ridiculous 89) being assisted solely by clerks who are largely in their late 20s: the senile being aided by the puerile.
Consider the futility of relying on clerks for a complicated topic like testing in the Ricci case. Do you think Justice Ginsburg's clerks were told the truth about testing when they were in law school? I don't care what your LSAT score is, to understand the reality behind Ricci, you have to do a lot of self-education and you have to learn about how the world really works. And that takes time. I moved to Chicago at age 23, and from then on I heard a lot about fireman and policeman testing, but it took me until my mid-30s to develop a mature understanding of the subject that wasn't just based on idealistic assumptions about how things should work. And I'm still figuring out things about fireman testing that make me say, like Huxley reading The Origin of Species, "How stupid of me not to have thought of that."
Occasionally, we see Justices instead hiring grown-up clerks with some experience of life (Justice Thomas recently hired a clerk who had already made partner at her law firm), but the salary is only around $65,000. (Supreme Court clerks get big signing bonuses from the private law firms that hire them when their year is up, but still ...)
What we need is a modest budget (say, $3 million per year across the 9 Justices) to allow each member of the Supreme Court to hire a mature Chief of Staff to manage the clerks, with, say, a three year term.