What does the word "wind" mean in this John Donne poem?

Solution 1:

It means wind

It doesn’t have some other, lost meaning here (as far as we know): it refers to that annoying thing that’s always against you when cycling and refuses to make an appearance when you’re trying to fly a kite.

The pronunciation isn’t really much of a problem either, because in John Donne’s (which, as BoldBen points out in his comment, was also Shakespeare’s) day, the weather wind was quite unequivocally pronounced /waɪnd/,¹ with the same vowel as mind, behind, kind, etc. The weather wind was completely homophonous with the curvy-twisty verb wind: they both had a long vowel.

So what happened?

Historically, the long (and by then diphthongised) vowel in this word came about as the result of homorganic lengthening about a thousand or so years ago and was completely regular. For some reason, though, there was a gradual change starting some time in the 1700s or so, whereby the vowel was shortened. Part of the OED’s etymological section on the word is worth repeating here:

The normal pronunciation would be /waɪnd/, as in behind, bind, find, grind, hind, mind, rind, etc., and this pronunciation remains dialectally and in ordinary poetical usage. The pronunciation /wɪnd/ became current in polite speech during the 18th cent.; it has been used occas. by poets, but the paucity of appropriate rhyming words (such as sinned, thinned, dinned) and the ‘thinness’ of the sound have been against its general use in verse. The short vowel of /wɪnd/ is presumably due to the influence of the derivatives windmill, windy, in which /ɪ/ is normal.²

The explanation as to why wind became shortened seems plausible enough: there were some derivatives, like windmill, which would regularly have developed short-vowel pronunciations (see the section Pre-cluster/polysyllabic shortening in the homorganic lengthening link above), and those could serve as the basis for a shortened version.³ After all, if it’s called a /wɪnd/-mill, surely it should run on /wɪnd/, not /waɪnd/. This is called analogical levelling and is very common.

But that’s backwards!

Plausible as it may seem, there’s a bit of a problem with the analogical levelling detailed by the OED in this particular case, and it is one of direction.

There are quite a few other words/roots that exhibited exactly the same circumstances (root had long vowel/diphthong, derivatives had short vowel), and which were at some point subject to a similar process of analogical levelling in order to generalise the same vowel throughout the root and its derivatives. In pretty much all of them (at least all the ones I can think of), however, the levelling went in the other direction: the derivative took on the long vowel from the root, rather than the root taking on the derivatives’ short vowel. For example, kind had its regular long vowel, but kindly (the adjective) ought to have had a short vowel (like another derivative, kindred). K/ɪ/ndly would make just as good a basis for k/aɪ/nd to shorten its vowel as w/ɪ/ndmill was for w/aɪ/nd. But instead the opposite happened with kind(ly) and lots of other words.

Typologically speaking, this is in fact rather what you’d expect. In most cases, the base word is likely to be more common than its derivatives (kind is certainly more common than kindly, and wind is definitely more common than windmill), and analogies normally involve the most common variant imposing itself upon the less common ones. With wind, it seems to have gone the other way.

So while it is perhaps not implausible in itself that wind was shortened because of derivatives with short vowels, it remains unexplained why the analogical levelling went this way with this particular word (or words, if you include windy), shortening the vowel in the root word; instead of going the other way and lengthening the vowel in the derivatives, as happened with pretty much all other comparable roots where levelling happened.

So I think we have to say that the shortening of the vowel in wind is ultimately unexplained.

Notes:

Or whatever exactly the pronunciation of the vowel was at that particular stage of the Great Vowel Shift. It was probably closer to [əi̯] or [əɪ] or something like that. I’ll just write /aɪ/ here to refer to the vowel that, at any rate, ended up being /aɪ/ in Modern English.
Note that this entry in the OED was written for the 1926 edition—I doubt you’ll find many poets these days who still cling to the old pronunciation of wind, and I can’t say I’ve ever heard of any dialects where it survives to this day, either.
Both the OED and the Etymonline article include the adjective windy in this group, but I think that is likely an error. In Old English, windy was windig, which doesn’t satisfy any of the three conditions for pre-cluster/polysyllabic shortening. Regularly, windy ‘pertaining to wind’ should have the same vowel as windy ‘following a curved or twisted course’.