Use of definite vs indefinite article is sometimes largely based on whether the speaker assumes his listener knows the meaning of 'reeling gait.' If this is the case here, such as in the first example you give (everyone knows the kind of gait I'm talking about or everyone knows the gait of a drunken man) she might very use 'the reeling gait.' This is true whether or not the speaker's assumption is valid.
Alternatively, if the speaker is about to explain what the meaning of 'reeling gait' is, the use of 'the' is also highly idiomatic.
Edit: in general, if the meaning of gait is known from the context of the text or the discourse between speaker and listener (either of which would include the speaker demonstrating the type of gait she means), the speaker will be apt to use 'the'.
Another example: go into my car and look into the glove compartment for the gloves I bought as a Christmas gift at JCPenney.
The context is that it is assumed everyone knows that cars have glove compartments, so the speaker says 'the' and the speaker assumes you will know which gloves he means by 'the' gloves, even though the listener has heretofore been entirely ignorant of their existence. Either there is only one pair there, or there are several pairs there but only one pair can reasonably be identified in the manner the speaker has described them. If the speaker has forgotten that she has bought two pair of gloves recently and they both could be taken to mean 'the' gloves, the listener might a) choose one pair, b) choose both pairs, c) choose none and ask for clarification. Edit: this is now meaning in discourse, and why questions asking you to plug in the "correct" article in a standalone sentence are misguided. Because for everyone of those questions, I can almost assuredly come up with a context when either the indefinite article or the definite article may be appropriate. Sorry, I got on a soapbox about the many quizzes like this one, even though that one is one of the better ones I've seen.
Note that I have not said anything about first mention or second mention of an item. This kind of explanation is an oversimplification and anyway is not true that speakers will not reference the same item with an indefinite article in a second mention of it.