How and in what way did the Danes come to influence English?

The source you cite seems to confuse two different sources of Danish influence in the English Language: the Jutes and the Danes.

• The Jutes are one of the peoples who invaded Britain from 449 onwards, along with the Angles, the Saxons and the Frisians.
• The Danes came as a second wave 4 centuries later (roughly from 850 to 878) and, as shown in @Robusto's answer, carved themselves the kingdom named the Danelaw.

The confusion comes from the fact that modern Denmark and Jutland are today the same place. However it is not certain that the Jutes came from modern Jutland. Let me quote an excerpt of the book "Origin of the English Language - A social and Linguistic History" [p. 53]:

"So it was first thought that the Jutes came from what we now call Jutland; the Angles from the western side of the Jutish peninsula and the east bank of the Elbe; the Saxons from the Elbe to perhaps the mouth of the Rhine. More recent Archaeological evidence locates the Angles farther south-east and the Jutes on the coast, near the Frisian islands off the coast of Germany and the Netherlands"

As a matter of fact, the real identity of the Jutes and their place of origin cannot be reliably established just from ancient texts because they contradict each other. Venerable Bede for instance, often cited as a source lived more than 250 years after the Anglo-Saxon migrations. If you hypothesise that the Jutes migrated to Jutland after the 5th century then the Wikipedia article stating that "many Danish speakers are able to understand some spoken Frisian" makes perfect sense. After all, people migration and splitting was quite common in these times. Consider for instance the migration of Goths, splitting and then travelling to Spain, Italy and Russia or that of the Vandals: to Sicily through Spain, Morocco and Tunisia.

Furthermore, as you have rightly pointed out, the distance between the Frisian islands and England is much shorter than the distance between modern Jutland and England. The most convincing observation is that the Jutes are said to have landed in Kent... That is quite possible if they came from Friesland but less likely if they came from Jutland. The Danelaw does lie "in front" of Denmark but Kent surely doesn't.

I do not mean to underestimate the contribution of Old Norse to English; it is instead, well established. One often cites for instance many words in "sk": (sky, score, skirt, skill, scab, scale, scrap), or such important words as get, die, call, egg, raise, take. Nevertheless one has to be careful when ascribing the etymology of an English word to Danes. It could be from Danish influence or it could be of more ancient Anglo-Saxon origin.

As for ta and thank, the origin of thank is believed to be Proto Germanic (thankojan) and was already present in Anglo Saxon English before the Danish invasions. However, the Old English (þancian) and Old Norse (þakka) versions followed parallel evolution paths and met again when Danes landed in Britain. So yes "ta" looks like Old Norse indeed but it caught up because it met an already well established cognate.

It must be remembered that the Saxon king Ælfred the Great (A.D. 849-899) presided over the southwestern part of England in his time (little beyond what were then known as West Seaxe and Suth Seaxe), and that for several hundred years the Danes had the rest. Here is a map (c. 10th century) of the Danelaw (the area of Danish settlement and dominant influence).

In fact, the Norman Conquest was only the second conquest of England in the 11th century. The first was begun by Svein Forkbeard, the Danish king, and was completed by his son Cnut in 1016. King Cnut (also styled "the Great") ruled all of England for nearly 20 years. It is against this backdrop that we must consider the English language to have been profoundly influenced by the Nordic tongue.

A couple more points on the influence of Old Norse on English.

The various forms of the third-person plural personal pronoun, "they/them/their" are believed to come from Old Norse þeir/þaim/þeira. I'm sure there are many other words, but this shows at what a fundamental level the language was influenced.

Also, Danish/Norse influence helped to make English simpler. It was during that period of history that the language lost some of its more difficult grammatical features - in particular declensions, the complex system of word endings used to distinguish subject from object ("who did what to whom").

Old English used a horribly complicated system with over 300 endings (5 cases × 3 genders × 2 numbers × (7 types of noun + 6 types of adjective) = 312. Plus there were additional rules for personal pronouns, and some irregular nouns that didn't follow the usual rules.

It is thought that the influence of Old Norse helped to get rid of the Old English system of declensions. Modern English (and Middle English) instead use a much simpler system of word order and prepositions.

England before 1066 (and after) isn't one place.

Especially so; Northern England was very different historically and literally than the South. Until the Norman conquest everywhere north-east of roughly London-Chester was a Viking country not an Anglo-Saxon one. This is still obvious in the place names. See for example this link.