Well, it's not talking about a specific slug, actually. Slug here means a strong drink—"a slug of whiskey," for example. It's being used in a figurative sense in the sentence you quoted, of course.
Feeble means "lacking strength, weak, infirm" (Oxford English Dictionary). Petering out means "dwindling away to nothing" (The Phrase Finder).
To sum up: the economic recovery is lacking strength, and in fact is dwindling away to nothing. The only way to truly fix the economy is with an unpleasant but necessary period of austerity, in which we forgo luxuries and make do with the bare necessities of life.