How can banks break themselves up? If they refuse to how can the federal government force them to break up or break them up themselves?
This is an interesting question. The banks themselves will not naturally divide themselves. There are great returns to scale in banking, and risk can be greatly diversified by being massive and having more clients in different positions.
This means they would have to be broken up legally. So, one would need legal cause to break them up, of which at the moment I can think of none that seem to strongly apply. One would likely need a new law. Perhaps they are violating an anti-competitive clause, eg. they are monopolies. (This would require quite a radical new interpretation of the existing laws, though.)
If this is the case, the Federal Trade Commission and Department of Justice generally have purview over past monopoly-busting activity. They bring suit against the company in question and these suits can eventually lead to breaking up a firm.
Generally speaking, there is a strong disincentive to break up. The firms did not become large by accident. It improves shareholder value.
Assuming the banks were not breaking any laws, it would be considered a "taking" under federal law to try and break them up using some excuse under the Commerce Clause. In other words, the Federal Government would have to pay off the shareholders the present value of all lost future profits into infinity. They would lose economies of scale and be forced to take higher levels of risk. The shareholders have property rights.
The Congress would have to make a finding that large banks are intrinsically harmful to national commerce. The courts would likely require proof that the finding is true and not politically motivated. While courts give wide deference to the findings of the two political branches, this would impinge strongly on the property rights of existing shareholders.
The more likely path that would have a better chance of passing muster, would be to set a maximum level of concentration for competing banks for future mergers. Historic decisions should not be brought up for review, so if a bank has been given a monopoly in a specific market by combination, that is out of bounds for making any changes. On the other hand, if somebody founds a new bank in that market, the large bank could not buy it up.
We already have concentration ratios present in regulations but those ratios can change when a President decides to change them. Or, the government can simply choose not to enforce the law and in doing so allow a de facto change in the ratios.
Congress could write language that does not control bank size but does control bank concentration in various markets. Unfortunately, that is unlikely to work.
Congress would almost certainly put weasel words into the law allowing the executive to suspend the law in times of banking crises. Indeed, that is how many banks became large. The government wants to minimize the expenditure of funds through the FDIC and so allows buying banks to use their capital to capture a failed bank. In doing so, they allow purchases that would never be permitted under normal circumstances.
To get around that issue, part of the law would have to require Congress to raise taxes and pay higher bailout amounts to increase future competition. A nation of taxpayers is unlikely to want to see their taxes increased when someone else is eager to bail the public out.
I don't see a credible path to breaking up the banks.