I stated writing a timeline a while back which started with the election of 1860 going to the House. My PoD was different -- in mine, Lincoln won on election day as in OTL, but died of a heart attack before the electors voted. Republican electors split between Seward and Hamlin, throwing the election to the House.
As per the 12th amendment, the House votes by states among the top 3 electoral vote getters. The Senate votes by seat among the top 2 VP vote getters to pick the Vice President. A majority is required (18/34 states in the House, 35/68 Senators in the Senate). Note that under then-current law, the new President is elected by the old lame-duck Congress, not the newly elected Congress which doesn't convene until December 1861.
In my scenario, the Presidental contenders are Seward, Breckenridge, and Bell. On the first ballot, I give all Republicans to Seward; all Independents, Know-Nothings, Southern Oppositionists (ex-Whigs), and antislavery northern Democrats to Bell; and all southern Democrats and all proslavery northern Democrats to Breckenridge.
Seward wins 16 states: Connecticut, Indiana, Iowa, Kansas, Maine, Massachusetts, Michigan, Minnesota, New Hampshire, New Jersey, New York, Ohio, Pennsylvania, Rhode Island, Vermont, and Wisconson.
Breckenridge wins 9 states: Alabama, Arkansas, California (yes, both lame duck California congressmen were pro-slavery), Florida, Georgia, Louisiana, Mississippi, South Carolina, and Virginia.
Bell wins 5 states: Delaware, Illinois, Missouri, Oregon, and Tennessee.
Four states are deadlocked between Bell and Breckenridge: Kentucky, Maryland, North Carolina, and Texas.
Meanwhile, the Senate votes between the top two running mates: Hannibal Hamlin (Lincoln's running mate -- a Maine Republican) and Joseph Lane (Breckenridge's running mate -- a proslavery Oregon Democrat). Assuming a party-line vote, Lane wins easily (38-26 with two Know-Nothings who could break either way). Both Know-Nothings and five antislavery Northern Democrats would need to vote for Hamlin despite the threat of Southern secession in order for Lane not to win. There are only 7 Democratic Senators from free states in the 36th Congress, and at least three of them (Joseph Lane himself, William Gwin of California who IOTL was imprisoned as a Confederate sympathizer, and Jesse Bright of Indiana who IOTL was expelled from the Senate as a Confederate sympathizer) I can't see voting for a Republican under any circumstances. Lane has a minimum of 35 votes, which is a majority.
Lane is now the Vice President Elect, who presumably becomes President if the House stays deadlocked (the Consitution is ambiguous until the 20th amendment is ratified in 1933).
In the House, Seward is two states short of a majority. With Lane as VP-Elect, I can see two states flipping to Seward. Oregon has one Representative, Lansing Stout, who has already lost his bid for reelection because Lane pulled strings to prevent the Oregon Democratic Party from renominating him due to Stout's opposition to slavery; Stout would likely vote for Seward to keep Lane out. Likewise, at least two of Illinois's six Democratic representatives would probably switch their votes to Seward to break the deadlock.
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In your scenario, two things are different. First, Lincoln is still alive, so he's the Republican Presidential candidate. Second, Douglas got more electoral votes than Bell, so he's the third candidate in the House.
Lincoln would do the same as Seward on the first ballot. Douglas would probably be weaker on the first ballot than Bell, as the Constitutional Union Party is the result of a merger between the Know-Nothings and the Southern Oppositionists so Bell can count on their vote while Douglas would have to convince them he's better than Breckenridge even though Breckenridge finished ahead of Douglas in most (all?) of the South. Most of the deadlocked Bell/Breckenridge states would go to Breckenridge, and some of the Bell states might deadlock or even go to Breckenridge rather than going to Douglas.
The Senate vote is identical. Same candidates, same Senators, same outcome.
The second ballot in the House is where things get interesting. Stout is just as likely to vote for Lincoln to keep Lane out, but the flipping of Illinois is much less likely. As an Illinois Democrat himself, Douglas has a ton of leverage over Illinois Democratic Congressmen, leverage which Bell wouldn't have had. If Douglas wants to, he can probably deny Lincoln a majority for several votes in hopes of emerging as the compromise candidate.
I don't think he would actually emerge as the compromise candidate, though. Lincoln would only need to pick off one state from him in order to get a majority, while Douglas would need to pick up at least 13 states to get a majority. And with the firmly pro-slavery Lane as the VP-elect, it'd be very difficult for Douglas to pick off Breckenridge states, so most of his gains would need to be at Lincoln's expense. The most likely scenario is Lincoln winning on the bazillionth ballot.