Why is the 'w' silent in "sword"?

Appropriately it's silent in answer. I couldn't find any other root -sw- words with a silent w.

As for sword, I found this from H.L. Mencken's The American Language from 1921:

As for the consonants, the colonists seem to have resisted valiantly that tendency to slide over them which arose in England after the Restoration. Franklin, in 1768, still retained the sound of l in such words as would and should, a usage not met with in England after the year 1700. In the same way, according to Menner, the w in sword was sounded in America “for some time after Englishmen had abandoned it.”

(There's a whole host of silent w s at the start of words but not after s, but they can be considered another "family": wring wrap wrong wrist write wraith wreath wraparound wreck wrath wreak wreck wrench wreckage wrecker wrestle wren wriggle wretched wrest wrinkle wristlet wristwatch writ write writhe wrought wry.)

I have actually heard someone pronounce the /w/, and not facetiously. It must be rather rare, though, and I might say it's a hypercorrection. In any case, my guess is that [wɔ] and [wo] have a tendency to become [ɔː] and [oʊ] in the neighbourhood of consonants such as [s] that don't change much in the presence of labialisation. The matter is complicated a bit by the spread of non-rhotic accents.

The original word was /sweord/, which according to the usual Old English reconstructed pronunciation would be [sweort] or [sweʊrt]. I imagine [eʊ] dropped to [əʊ] or [əu], which is an allophone of [o]. By this time the [w] had become vestigial and was ultimately dropped, but English spelling is far more conservative than its pronunciation, so the /w/ was retained.

<w> is there because it was pronounced after the advent of printing press. So, the spelling stayed, but pronunciation changed. In three cases, andsƿarian > answer, sƿeord > sword, tƿa > two, <w> is there, but not pronounced.

On the other hand, <w> (or its OE <ƿ>) is not there in some words because they were not pronounced by the time printing took over: OE sƿilch > such, sƿa > so

Just a theory: As sword is a thing of knights and noblemen. The silent w may be due to Norman-French pronunciation habits and this pronunciation was generally accepted.

The w in spelling shows the origin of the word and its connection with German Schwert. With the w-spelling it is clear at once that the word does not belong to the word family "sort".

This answer on ELL has explained the reason why the W is silent in sword but pronounced in swore. That answer is rather long, but I'll simplify it here:

Between 900–1400 AD, there was a sound change through which a W was lost when it was preceded by a [s] or a [t] and followed by a back vowel [ɒ ɔ o ɑ]. [Wikipedia]

The W in two and sword is silent because of that reason (i.e. [t, s] [w] [back vowel]).

Swore, sworn and swollen also lost their W's at one point due to the same reason, but were later on restored due to analogy¹ with swear and swell. Swear and swell hadn't lost their W's because they had front vowels after the W. [Trask's Historical Linguistics]

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^{1. Analogical change is a type of language change in which some forms are deliberately changed merely to make them look more like other forms.}