The question explains itself I hope.
Mein Fuß ist eingeschlafen (* my foot slept in, fell asleep)
This would be said if you sat on your foot too long so that it becomes numb.
This is of course an absurd figure of speech1. Its derivation should be questionable.
From the little I know about the language, I suppose a genetic relation between Schlaf and schlaff would be difficult to account for. The etymology of Schlaf is relatively uncertain, without comparable cognates, incongruent with PIE *swep- and *ses- "sleep" that show up elsewhere in Indo-European languages. Likewise, the come about of geminates like -ff- is relatively uncertain. In any event, the comparison is not completely unreasonable, cp. e.g.: schleifen, -schliff-; streifen (i.e. glatt streifen, of bed sheats), straff, entirely uncertain Straf- (cf. Gk. strepho, versus Strophe, and Richterspruch, Tenor, etc., IMHO).
The semantics match well, so that the comparison surely has been drawn before. In extreme cases, a hand that has been layed on can be so numb that it hangs completely limb, whereas the less sever cases only result in a prickling sensation whereby the association to schlaff might not be apparent.
The pertinent question then is if the idiom followed only after sleep had been established, or if it was rather a parallel development from a common root that schlaff somehow diverged from for reasons that should elucidate schlaf- as well. The prefixing, einge-, should be considerable as well.
1: To address comments who disagree, I mean it is an anthrpomorphism in a sense. Although, there is no active form schläft, and the tired limb is not said to wake up either. Schlaf is also used as euphemism as the cousin of death²
2: Thinking about a closer relationship between sleep and corpses, i.e. German Leiche, I have to note the complicated correspondances that involve sanft, sacht, En. soft (by Nasal Spirant Law), Kraft, Nl. kraaft (?), lache, En. laughter (??), leicht, Lat. levis, also lupfen, En. to lift, maybe lüften (of skirts?), and of course legen, liegen, sich hinlegen (substandard palatilized leng, cp. also sich lang machen).