WI Contingent Election in 1960?

Suppose that the 1960 presidential election goes further down the wire, and the Kennedy-Johnson ticket holds a plurality of the electoral and popular vote, but falls short of the magic number of 268 electors. Harry Byrd and his unpledged electors make known their demands regarding segregation, but Jack refuses to sacrifice his civil rights plank. The country stares down deadlock that may be resolved by a contingent election.

It looks like there are many ways the election could proceed. If the electoral chips fall so that the unpledged electors can be a kingmaker to either ticket, would Nixon be willing to play ball with Byrd? Else, is LBJ's presence enough to get the Southern Congressional delegations to line up for JFK?
 
JFK would win in the House. The new House had 29 delegations with an outright Democratic majority (I'm not even including tied delegations). https://en.wikipedia.org/wiki/87th_United_States_Congress (It is rather startling today to realize that both Representatives from Utah were Democrats, as were both Representatives from Idaho!) The only southern Democratic delegations which I see as insisting on voting for Byrd are MS and AL.
 
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