The trick with such translations is not to translate literally, as that often results in a rhythmically awkward or unwieldy expression, but to craft a suitably poetic equivalent.
Havah means “Let us”; nagilah is a rather old-fashioned and literary way of saying “we shall rejoice”—together, they mean “Let us rejoice.”
An excellent question, because the agonizing and linguistic contortions surrounding this word among non-Hebrew speakers have always puzzled me.
The traditional translation—lovingkindness—is totally inapt on several grounds: it’s a made-up word, cloyingly sentimental, semantically wrong, and rhythmically horrible, wreaking havoc on the meter of any verse in which it is present.
Absolutely—especially if the standard modern Israeli pronunciation is involved, whereby many letters (such aleph and ayin; tet and tav; het and khaph; kaph and quph; shin and samekh) sound alike that in the traditional Sephardi or Yemenite pronunciation, do not.
To demonstrate this—and the utter failure of conventional, quasi-phonetic transliteration of Hebrew in Roman characters to maintain the distinctions in Square Hebrew script between such homophones—see my poem, Modern-day Ecclesiastes, which begins:
The short answer is that you fairly quickly develop an intuitive sense, from hearing (or reading) how other people use that verb. But if you’re looking for a broad rule of thumb to get you going, here it is:
A straightforward, modern Hebrew rendition would be something like
זעמו של אלוהים יעניש את נשמתך