As he shared that balance of power with four other independents, it's not really accurate to claim that he alone controlled it. Moreover, an internecine fight among Republicans meant meant that even if all five had cast their ballots for Minority Leader James Mann, Champ Clark would have been Speaker regardless.

The two words that best describe the party organization of the 65th Congress are unadulterated chaos. From the best political study of that era, Livermore's Politics is Adjourned:

For the first time in history, elections for the House ended in a dead heat between the two major parties. Each had 215 seats, and the balance of power lay in the hands of 5 minor-party members: a Socialist, an Independent, a Prohibitionist, and two Progressives.

Besides London, these were (in order of the Livermore description): Alvan Fuller (IR-MA), Charles Randall (PROH-CA), Melville Clyde Kelly (PROG-PA), and Whitmell P. Martin (PROG-LA). But their party alignments barely begin to scratch the political mess that was the 1916-1918 War Congress.

The closest comparison might be the legendary 36th and 37th Congresses at the advent of the Civil War. Unlike those Congresses, which at least had as their party-transcending political axis the question of the Union and what measures (if any) that they'd support to preserve it, the 65th was just all over the place. Interventionist versus isolationism were certainly big issues in many Congressional races in November along with the related ones of draft versus volunteer army, preparedness, and the real lightning rod of the now-completely forgotten McLemore amendment, which would have prohibited Americans from traveling on armed merchant ships. To give you an idea of just how much party lines got blurred during in that time frame, the last was proposed by a renegade Democrat, violently opposed by the White House, received 3/4 of its support from Republicans, and probably cost its supporters more seats than any other issue in 1916 (and given far more Republicans than Democrats voted for it, it was a major reason why the latter broke even in the House.)

But wet versus dry mattered a lot in many races (this was the election that finally provided enough votes to pass Prohibition), as did progressive (in the Bull Moose sense) versus conservative, and sectional (the bloody shirt was waved in a lot in the North against Wilson's caving in on race to Southern Democrats) - along with sheer partisanship since Woodrow Wilson wouldn't appoint Republicans and in turn Teddy Roosevelt felt Wilson was incompetent and probably the worst President since Buchanan. The cacophony of issues run on in November 1916 resulted in perhaps the most ideologically scattershot Congress in history, the likes of which really hasn't ever been seen before or since.

Despite this, some historians like London's biographer Gordon Goldberg have bought into the theory (espoused in multiple newspapers at the time) that had the five independents all been socialists - not impossible given close election results in various districts - that a revolution could have taken place in Congress if they'd required concessions for their votes. This mindset is likely where the inaccurate statements in Wikipedia and elsewhere come from about London controlling the balance of power.

Given the above balkanization, though, that was extremely unlikely to happen even if all of them had been elected, but during the lame duck session in December, something else occurred that made all of this moot: James Mann decided that Republicans might offer Woodrow Wilson some support in his attempts to try to end the bloodshed. This did not go over well with his caucus. From Livermore again:

(I)n connection with the peace negotiations in which Wilson became involved as a result of the German peace note of December 12, Wilson's reply on December 18th ask(ed) the belligerents to state their peace terms (and precipitated) an angry controversy. Senate Republicans blocked a resolution calling for a blanket endorsement of the note. In the House a similar resolution caused a violent explosion in the Republican ranks when Minority Leader James R. Mann unexpectedly arose in defense of Wilson as a peacemaker and urged bipartisan support for his endeavors. Die-hard interventionists of the Roosevelt persuasion hotly denounced Mann as a tool of the Kaiser and challenged his moral right to the continued leadership of the party.

Mann's loss of the TR interventionists thus meant that Champ Clark was going to remain Speaker with or without the vote of Meyer London and the other independents. The vote was 217-205 for Clark on the first ballot in April 1917 with all independents besides Fuller supporting Clark, but more notably 7 other Republicans either abstaining or supporting other candidates (including Thomas Schall outright voting for Clark, which is one of the few times a MoC has crossed party lines on the Speaker vote without significant later consequences within their caucus - he served 3 more terms in the House as a Republican and was later elected to the Senate.) Even if Mann had shown a genuine chance of winning, plenty more TR disciples would have been available to sit out the vote and sabotage him. The TR interventionist faction numbered far more than the 7 who recorded their protest votes, and indeed that faction of the party had already done so on a grander scale with their tepid embrace of Charles Evan Hughes, a stance which was a major factor in costing him the election.

In terms of largess granted to the independents, none of them received committee chairs, the traditional payout for independents supporting a leadership vote, and it doesn't seem as if London received anything whatsoever for his vote. In turn, he went on to vote consistently against Wilson, being one of only 7 members of Congress to score a zero in supporting Wilson's 'acid test' votes per the National Security League. (As a reflection of the scattershot ideological nature of the 65th's composition, 5 of the 6 others were Democrats.) This and other things - for instance, a Socialist candidate was partially blamed for a Democratic loss in an important Senate by-election in Wisconsin - provoked enough anger within party leadership to agree to run a Democratic-Republican fusion candidate against him in 1918. He lost that election before regaining his seat in 1920, lost it again in 1922, and then died tragically in a traffic accident in 1926.

Of the remainder of the independents, probably the biggest impact of all of them came from Randall's authoring the War Prohibition Act and placing it as a rider onto a food control bill (a bad harvest, Allied food requirements, and speculation by farmers had created a grain shortage) which prohibited food from being distilled into alcohol for the duration of the war. This essentially created a dry run for Prohibition in 1917 and 1918 before the Eighteenth Amendment took effect in 1920. However, as the bill passed 365-5, there's no evidence that leadership allowed Randall to do so as payback for his vote.

Within a few years, Fuller and Kelly had returned to the Republican party (with the former notoriously serving as governor of Massachusetts during the Sacco-Vanzetti trials), Randall had lost the Republican part of his fusion status, lost his seat to a Republican and returned to local Los Angeles politics, and Martin served the remainder of his terms as a Democrat.