Seems like you've already identified the major root causes for such a ramshackle historical response to the Mongol incursions by Christian Europe. My understanding is that the Battles of Legnica (Poland) and Mohi (Hungary) weren't even supposed to happen according to the European strategy, the actual plan was for the Polish & Hungarians (and also the Bohemians) to combine their forces in order to have a much better chance against the Mongols - it just got pre-empted by Subutai splitting off a division to crush the Poles and demoralize the Bohemians into retreating with hardly a fight, after which the main Mongol army was free to pulverize the Hungarians.
In order to affect a European victory, it'd probably be easiest just to have Legnica happen differently or not happen at all. The
Bohemian king Wenceslaus was apparently literally just 1-2 days away from the army of the Polish dukes & cities led by his brother-in-law
Henry 'the Pious' of Silesia when the Mongols got the jump on the latter; have the Mongols be delayed just a little bit and the Polish-Czech armies could've combined as planned. The Bohemians were said to have 2x as men as the Polish did, making for a force large enough that Subutai's lieutenants may not have risked an open battle with them at all (it is worth noting that when they did harass Bohemia after destroying the Polish army, Wenceslaus defeated them).
After that it's a matter of this combined host marching to the relief of the Hungarians while inevitably still enduring harassment by the Mongol secondary division trying to slow them down. Mohi was actually a pretty hard-fought affair where Batu Khan wanted to retreat at one point and the Poles/Bohemians showing up at the right time would seriously screw with the Mongols. That is, assuming Subutai (who, it must be conceded, was a military genius - probably the greatest one of the medieval period - and who inspired modern generals as Hannibal did) and his lieutenants even bother risking a pitched battle at all against the allied armies under these circumstances.
Above all the Europeans' main win condition, so to speak, would just be to hang on until Ogedei Khan dies (better pray he dies on schedule and his life doesn't get extended by the butterfly effect, fortunately for everyone who isn't a Mongol he was aging and a huge alcoholic) and the Mongols in Europe go home to deal with the question of succession. Just a few more months, basically; if Bela IV, Wenceslaus & Henry can do that, they're good. They will have time & less ruined bases from which to rebuild, absorb the lessons of this first round and prepare for any future Mongol invasion, which was exactly how
Bela IV's grandson defeated such an invasion by the Golden Horde ~40 years later.
As for the rest, England & France shouldn't be relevant to any response to the Mongol invasions. If they have become relevant, then that must mean the Mongols are tearing up Central Europe and the situation for Christendom has become very grim indeed.
I don't think there's any helping the Papal-Imperial/Guelph-Ghibelline conflict at, or even for some decades before, the mid-13th century point when the Mongols struck: the Hohenstaufens had been locked in intense conflict with the Papacy since Barbarossa's day. On one hand the Papacy had reached the peak of its medieval power & influence (this is not long after the papacy of Innocent III,
one of the greatest champions of the idea of Papal supremacy not only in spiritual matters but also over Europe's temporal princes, and still some decades before the disastrous 'Babylonian Captivity' in Avignon & following schism which severely diminished Papal authority).
On the other hand the Staufer are being led by their most brilliant and most ambitious Emperor,
Frederick II 'Stupor Mundi'/'Wonder of the World', and he poses a mortal threat to the aforementioned Papal dreams of both spiritual & temporal supremacy - he's out to further build up & throw around the temporal power of the Emperors, and has also inherited the Norman Kingdom of Sicily which gives him a huge & wealthy powerbase in the Popes' underbelly. The Polish high duke Henry gained an advantage over his rivals in the fragmented Poland by allying with the Papacy against Frederick, IIRC that's at least one of the reasons as to how he gained the nickname 'The Pious'. Anyway, point being, it would probably be best for the Christians of Central Europe if they're able to hold the Mongol steamroller back without having to rely overmuch on either Rome or the HRE. If Mohi is won by the Christians or avoided entirely, you would also be able to avoid oddball opportunism like Frederick of Austria's attacks on the Hungarian western frontier.
You'll be pleased to know that the greatest beneficiary of the Mongol invasion of Central Europe being turned back will almost certainly be Poland. Bohemia more or less made it out OK, and Hungary was able to rebuild, but Poland got set back pretty hard by the death of Henry at Legnica as he had been a very capable leader & nearly fully reunited Poland after it fragmented in the 1130s. Even worse, Henry's branch of the House of Piast was from Silesia, and his demise + the incompetence of his successors doomed that part of Poland to further collapse & decline until Silesia was lost to the Bohemians & later Germans, not to be recovered (for the most part) by any Polish state until the modern period.
Henry surviving and going on to finish uniting Poland would produce a stronger, more western-oriented country that also includes Silesia with its wealthy towns, mines, etc. Probably this Poland will be more heavily tied to the states of the HRE, even if they continue to support the Popes against its Emperors; IIRC the Silesian Piasts intermarried frequently with HRE nobility/royalty, with Henry himself being half-German and married to a Bohemian princess. Decent chance that with more involvement in shenanigans to the west, this Poland won't have the time or energy to spare on eastern adventures and thus never unite with Lithuania, remaining a smaller but more compact and (with the benefit of Silesia's resources & deeper ties to the HRE's constituent states & statelets) probably ultimately wealthier kingdom.
Anyway, I don't think a Piast Holy Roman Emperor is necessarily going to happen but with such ties to the Premyslids, King Henry or his descendants might lend a much stronger hand to their Czech ally and (assuming contenders for the imperial crown still more or less look the same when the Staufer lose their grip, quite possible at least in Bohemia's case because Wenceslaus' successor was
the highly capable 'Iron and Golden King') derail the rise of the Habsburgs in favor of said Premyslids at the decisive
Battle of the Marchfeld. That would definitely alter Europe's future so massively as to make it unrecognizable.
