Great question! Just as a heads up, as I try to answer I’ll mostly be pulling from Barlow’s Feudal Kingdom of England and Stenton’s Anglo Saxon England. Both are great foundational books that give a feast of detail about the curious formation and qualities of the early medieval English kingdom.

As far as it pertains to their ability to appraise and collect a land tax (and that’s what the Danegeld was - an appraisal of land and it’s agricultural productive ability and a tax assessed thereon) we need to discuss the curious division of land and political authority that predates even the uniting of the Heptarchy, the 7 traditional kingdoms of the Anglo Saxons. There were significant differences between the forms of land tenure of various regions of England that would last well into the early modern period at least on paper, but they also bear some similarities.

In the heartlands of the kingdom that would come to finally unite the English polities, Wessex, the primary unit of land tenure was the hide. A hide was an area of land assessed to be enough for a family to live off the agricultural produce from. You’ll notice that it differs from our own modern units of land measurement - generally acres or hectares etc, in that it wasn’t necessarily contingent on area of land but rather its productive ability. This will actually prove an advantage later on. There were, as I mentioned, various other forms of land measurement such as the lathe, but they would fulfil the same purpose generally.

When grouped together, 100(ish) hides would form a unit known as the - yep you guessed it - Hundred. A Hundred was a crucial unit of local governance and relations, as the Hundred Court provided the lowest local level of judicial review and interaction for the vast majority of the population. Some historians have postulated that the Hundred was originally a unit of land that could be expected to provide 100 armed men in the case of conflict (which makes a deal of sense considering it would have around 100 families living on the hides within it). We rarely have exact measurement of how many troops these areas could provide but it’s a useful outline nonetheless. There were also various names for similar divisions of land (my favorite being the Wapentake). Hundred courts and this division of land tenure would survive well into the later medieval ages.

Containing several hundreds and based roughly upon the tribal groups of Angles, Saxons and Jutes that settled/invaded/mingled into Eastern Britain during the 6th and 7th centuries, you have the Shires. Some of these Shires were in fact former kingdoms of their own (Kent and Sussex for example) or formed smaller parts of already existing kingdoms (Devon, Wiltshire). Shires generally were overseen by a noble from the area known as an Ealdorman in the pre-Viking period and increasingly as Earls (etymologically related to the Norse term for their leaders, Jarls) who was ostensibly appointed by the King. And this part is crucial - as the Kings of Wessex united England under their banner over the course of the 9th and 10th centuries, they retained their ability to appoint and sometimes recall their powerful nobles from their positions as Earls. This kept them nominally as appointed officials of the King as opposed to functionally independent warlords - a phenomenon that was simultaneously plaguing France during this time, for example.

The other uniquely important official that kept the kingdom running was the reeve. Reeves were either local or increasingly royal officials who played an important role in giving justice, settling disputes and collecting revenue throughout the Anglo Saxon kingdom. How many reeves there were or their exact role(s) is murky until later in the Norman period, but their attachment to the Shire would eventually lead to them becoming known as Shire-Reeves, which would eventually be etymologically transformed into our term Sheriff. Neat!

All of this combined to give the Anglo Saxon Kings an exceptionally centralised and organized state apparatus, especially compared to other contemporaneous medieval polities. One has only to look at Ireland, Scotland, Wales, France or Christian Spain to see their fractured political organization. France in particular was plagued by disunity that the Capetians would spend the better part of half a millennia trying to amend. Some of this may be seen as a result of the need for strongly centralised response to Viking raids and invasions. The Burghal Hidage, for example, provides insight into a brilliantly organized system for maintaining and garrisoning fortified towns throughout Anglo Saxon England that also hints at well maintained records of local land tenure and populations. The Burgh system was intentionally built to provide Defense against Viking raids but also as an offensive tool as the Saxons began pushing into Danish lands in the 10th century.

This is all to say that when Danish raids and military campaigns came roaring back in the early 11th century and Aethelred the Unready (a great example of medieval pun humour) proved incapable of fending them off militarily, he had at his disposal a superbly organized (for the time) system with which to assess and tax his population that would have been the envy of any of his counterparts on the continent (besides as you mentioned Muslim controlled Spain or the Byzantines). The English King could rely on a network of reeves who could circulate the shires and their constituent hundreds (already hypothetically made up of relatively uniform units of land) and collect the necessary taxes to pay off the Danes. All this was overseen by Earls who were less independent of the King’s oversight than say a Norman Duke or Angevin Count, and could be relatively trusted to make sure the right amount of geld got to the right place.

In fact, especially after repeated payments of Danegeld proved to just be incentivising more raids and invasions by the Danes, English Kings would turn to spending this geld instead on hiring their own forces of North Sea mercenaries - groups that Edward the Confessor would notably finally disperse in the lead up to the Norman Conquest

What’s more, the Normans themselves would continue to collect geld taxes long after William shattered the English forces at Hastings. English taxes would form a crucial part of English King’s revenue throughout Norman and Angevin periods, and would allow them to compete (quite successfully) with the Capetian kings of France. But it should also be noted that the Normans brought with them very different forms of land tenure built around the manor and honour that would partially dismember the Anglo Saxon system - though elements of it are still visible today.

I hope this helped answer your question. I also have a previous answer where I mention the revenue collection systems of Medieval English kings so I’ll paste that here when I have time.