The 21-year-old full-back will spend the rest of the season at the City Ground and will join up with his new team-mates on Thursday.
The £250,000 prize was awarded to Forest last month when fan Shaun Lander won the Sky Bet random draw at Wembley.
Forest manager Stuart Pearce said: "Todd is a player we know well and we are all looking forward to working with him.
"He will be a huge asset to the team and I'd like to thank Shaun Lander for participating in the Sky Bet Transfer Fund and helping the club to sign Todd."
Lander, a landscape gardener who also won £5,000, said: "As a fan, I'm absolutely delighted I've played my part in helping to bring Todd Kane to the club - and all just by placing a £1 bet.
"Hopefully he can help our push for a play-off place."
Former England youth international Kane has yet to appear for Chelsea but has had previous loan spells at Preston and Blackburn and made seven appearances for Bristol City earlier this term.
Forest are currently subject to the Football League's Financial Fair Play embargo, but clubs are still permitted to register new players if they have 24 or fewer established players (players aged 21 or over that have made at least five starting appearances for the club) but only if the annual employee costs of the player being signed are less than £600,000.
Source : PA