If I have all six orbital elements, what equations allow me to calculate the period of the object?
Perhaps  is useful to you.
Re: Calculating the period of an orbit (for a meteor)
Yes, that shows a number of choices, but opens up a new question. The object is part of a meteoroid that has struck the earth's atmosphere. What's it orbiting? (I'm assuming this is debris from a comet. I suppose though it could be a full blown asteroid that skips off the atmosphere.) Not necessarily the sun, perhaps. I've heard of asteroids that have come inside the earth's orbit and gone back into space, possibly to revisit the area again, if not on a hyperbolic or parabolic orbit. The equations given on the web page show a need for the mass of the central body. What if there's no central body? On the other hand, bodies orbiting without a central body (or two) seems quite odd.
You say you have a specific set of orbital elements? Such a set must be in reference to a primary body. For asteroids, comets and similar objects, that primary body is almost always the sun, in which case the orbital period is the period for one orbit around the Sun.
If the object in question make a near pass of Earth its actual path through space can be modeled (or approximated) with three different Keplian orbits, each with possibly different orbital elements and primary body: 1) an orbit around the Sun valid for (long) before the Earth passage, 2) an orbit with Earth as primary for time around the Earth passage part, and 3) an orbit around the Sun valid (long) after the passage. Note that the Earth orbit segment will be a so-called hyperbolic orbit (as opposed to an elliptic orbit), meaning that the orbit is not closed (perodic) and that it therefore does not make direct sense to talk about an "orbital period". Each of the first and last orbit (those with the Sun as primary body) may or may not be closed, but for each that are closed you can then talk about orbital period around the Sun for that orbit.
If this is not helpful to you, perhaps you could state exactly what data you have and what you want to obtain?
The attached article somewhat explains what I'm after. It's from an article on meteor orbits. Eq. 53. As far as I can tell, he does not show how to get the mean anomaly, mu. This seems to be close to what I see coded into a program I have. It is output as "orbital period", an idea that may be stretched a bit. The eq here doesn't seem like it's producing the orbital period, but something shorter, DT.
I need to go back to the code to see if this is really a match. I'll return to this later.
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