The best thing to do is contact the original author and request that he or she update the repository to include a proper license file and copyright notice, along with license headers and copyright notices in each file.
If the author is totally unreachable, then you're on shaky ground, and I'd say the safest thing you can do is not use the code at all. The primary risk, in my view, is the possibility that the code might not actually be MIT-licensed, owing to the absence of a proper license declaration.
However, if you must use this code and you're sure that:
- the original author1 did not include a license file with the code, but
- the original author claimed that it was MIT-licensed in some authoritative way, and furthermore
- the original author is the only author of all the work in the codebase (or you can correctly identify all authors and/or copyright holders)
then your best course of action might be to add those things (license, headers, notices) yourself in your distribution of the code. Your use of the code under an MIT license would be generally defensible under the legal principle of estoppel ("the legal principle that bars a party from denying or alleging a certain fact owing to that party's previous conduct, allegation, or denial"). Because the author made a previous claim that the code was MIT-licensed, the author cannot pursue legal action based on a premise contrary to that claim.
Of course, all this is hardly sound legal advice -- I'm not a lawyer, and I present this information only to demonstrate how one might begin to formulate a proper legal defense for reusing the code. It seems like reasonably strong justification to my layman legal sensibilities, but software is a popular domain for niche regulations, so there may be jurisdictional and international concerns that I have not addressed. As always, your safest course of action is to consult a lawyer.
1 I place emphasis on original author to eliminate from my scope of argument the possibility that this is a re-hosted copy of the code from which the re-hoster lazily dropped the license file.