I will look into his memoirs about this. But.
I know there were considerable amount of reinforcement in the Highlands, and maybe even supplies. The towns were traditionally rich (originally Saxon - liek Zipsers - and Swabian settlers founded mining towns) and important industrial centers. At first Görgey wanted to break the siege of Lipótvár which was held by 1500 patriot against Simunich's 5000 men, from there they could have reached Pozsony easily (earlier the seat of the Diet, due to it's closeness to Vienna), or move to either Komárom or Győr, all three cases would landed them behind the imperials and on their supply lines.
The imperals also could hope for some popular support in the Highlands (or Upper Hungary), among the tót part of the Northern Hungarians those who speak Czech funny
a priest named Hurban recruited irregular force against the Revolution this movement was way less considerable than today the Slavic nationalists of Northern Hungary try to imply, the main body of the tóts sided with the Hungarian revolutionary government
So Görgey had to be kept in check. Windisch-Grätz sent a force from his troops, which made his army weaker. He also wanted to concentrate more units onto the Great Plain, most notably Schlick's, but at that moment when Görgey left, Simunich's too. I read that the imperial high command didn't have reliable intel nor about the strength of our forces neither their exact location this made W-G cautious (maybe even confused).>And which other noble house could take the throne?
I dunno if they had any plans on that, or who wanted to elect whom. In practice Kossuth became governor and weren't much talk about coronation.
Our kingdom without a king can work fine. The Holy Crown is the embodiment of the kingship, and the source of royal power. During the middle ages a tradition formed which is now called Holy Crown Doctrine (I'm not aware an official translation of the name, it contains/contained both customary and written law) which treats the Crown basically both as an abstract idea (with notions such as the will of the people) and as a legal person (such as the owner of the crown lands - this part can be familiar). From 1446 in the absence of a king (during interregnums for example) a governor (gubernator) gets elected and rules instead the king, the legal basis and the practice of this rule was regulated on ad hoc basis, there weren't explicit guidelines for that I think, but generally were under the supervision of the Diet/legislation (as the will of the people and the will of the Crown).
The kingless kingdom works as a republic with a "civilian" at the helm. In case of Kossuth, he was called a governor-president - which probably made many peeps twitchy, projecting a republican turn. Anyway keeping the kingdom in our case is a question of tradition and the continuation of a thousand year old legitimacy.
As a nice touch, the Prime Minister of the first "responsible" government as they called it responsible to the Parliament not the king
took an oath to the Holy Crown. They took it seriously.
Picrels: first and last governors.