Unfortunately, we just don't know. There is no clear Greek etymology for it, not just the initial Lake- part, but the whole word. This is from Beekes' etymological dictionary:
Λακεδαίμων, -ονος [f.] town and country on the river Eurotas (ll.). <PG?>
.DIAL Myc. ra-ke-da-mi-ni-jo /Lakedaimnios/ and [ra-]ke-da-mo-ni-jo /Lakedaimonios/, cf. Lejeune RPh. 68 (1994): 165-168.
...
ETYM The meaning of the appellative is unknown, so the name remains without etymology. Several proposals: a) to λακεδάμα [/lakedama/]...'bitter water [poured over groats] which the [Macedonian] countrymen drink' (H.); b) the second member is δαίμων /daimon/ in the sense 'part' (BechteI 1921, 2: 370); c) dissimilated from *Λακεν-αίμων /Laken-aimon/, to > Λάκων /Lakon/ and another ethnonym Αἵμων /Aimon/ (Szemerenyi Glotta 38 (1960): 14ff. with
ample discussion).
Acc. to Fick 1905: 90, however, it is rather a Hellenisation of a Pre-Greek word. Frisk thinks that Λάκων is a shortened form of Λακε-δαιμόνιος /Lake-daemonios/; the element Λακε- /Lake/ is found in Λάκε-θεν /Lake-then/, a deme of Eretria.
- The transcriptions of Greek words are my own and follows the same type as Beekes' does for Mycenaean words. The "PG" in the beginning means "pre-Greek."
Under Λάκων, Beekes has some further information:
ETYM Acc. to Dittenberger Herm. 41 (1906): 196, hypocoristic for official Λακεδαιμόνιος, which is why the fem. Λακαίνα /lakaina/ (for Λακεδαιμόνια /Lakedaimonia/ is found almost everywhere. Krahe IF 57 (1940): 119 supposes the name to be Illyrian and connects Lacinium, a promontory in southern Italy, Iuno Lacinia.
Fur.: 171117 thinks that the suffix -αινα is Pre-Greek.
So as you can see, there are many guesses, but no one knows for sure. It's doubtful that -daimon itself is even related to the Greek word daimon, and if so in what sense it is being used.