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Suppose that opinions in the UK parliament become so fragmented that no-one can form a government. What powers does the reigning monarch have?

At the moment Queen Elizabeth II is the Commander-in-chief of the British Armed Forces.

Could she she use this or any of her powers to rule until the politicians have sorted themselves out?

No.

This question is normally taken to mean "what powers does the natural body of the queen have in the UK constitution". The answer is "none". She must follow the advice of her Ministers.

It is not clear what you call "disarray". If you mean political parties splitting up, repeated votes of confidence, mass resignations form Cabinet. I.e. politics continuing in disarray, then no the Queen has no reserve powers. She acts only according to the advice of her Ministers.

It is absolutely unconstitutional for the Queen to rule as an absolute monarch "until the politicians have sorted it out."

If you mean the much much less likely situation of an illegal and unconstitutional government (say, for example, the Prime Minister ordering the army to round up and execute the opposition party). Then we are dealing with a situation that is beyond "what powers does the Queen have" and to a situation where "who will the people with guns chose to obey". This is by definition not covered by the constitution.

Questions about the power of the Queen come up occasionally on this site. The normal answer goes something like this. The queen has absolute authority to do as she pleases however she does not choose to use this authority and instead carries out the will of parliament. If the queen chose to remove parliament and rule in its place this could lead to some serious consequences for the queen and a possible revolt thus the queen so far has not chosen to use these powers and it seems somewhat unlikely that this will change.

Additionally I would question to what extent parliament is in disarray. While there is some difficulty with respect to brexit there is no suggestion that the parties are struggling (more than usual) with the other issues.

The risk of no-one being able to form a government is somewhat limited if there was a serious risk of this a no confidence vote would have been called by the leader of the opposition. If such a vote took place to conservatives along with the DUP would eventually express confidence in the prime minister or some other acceptable MP to avoid a general election.

The UK hasn't had a revolution or a new constitution recently to take the monarch's royal prerogative away, but custom puts most of her powers into the hands of HM government and the parliament.

So what we would be talking about is the monarch breaking centuries-old custom to rule directly. Right now I would bet that the vast majority of citizens, officials, and soldiers would not accept direct role, even if they are loyal to the Queen in her current role.