Kennedy would have won in the House if the votes had been fairly counted. Kennedy obviously won by fraud, Daley had no problem generating the +4000 votes Kennedy won by. But Kennedy also won MO by only 4000, and MO has a long and distinguished history (all the way up until at least 2000) of voting the Democratic dead and ghosts of St. Louis. If only real votes had been counted the result would likely have been Kennedy 263, Nixon 259 and Byrd 15 sending the election into the House where Kennedy would likely have won (Democrats held a 262-175 edge).
The House doesn't vote like that in case of no Electoral College winner. Each state gets one vote in the House.