The angles and saxons spoke a western germanic language, whereas modern danish is a northern germanic language.
EDIT: as a side node: Frisian is the language commonly regarded as being closest to English
die/der/das -> [see below] -> the
drei -> tre -> three
Donner -> torden -> thunder
Ding -> ting -> thing
daher -> derfor -> therefore
du -> du -> thou
I have no idea what happened to the definite article in Danish though: ein Haus -> et hus -> a house
das Haus -> huset -> the house
Aside: The Danish capitalization rules used to mirror the German rules until after WWII, where we switched to the Swedish rules.When you know Norwegian or Danish you can see a lot of similarities. The explanation I got was that around year 1000 Norway/Denmark was pretty much a naval superpower.
The union lasted until 1814, when Denmark (and by extension Norway) had supported Napoleon, and Sweden had joined the other side. Sweden got Norway as a "price". But during that period, a lot of people had started preparing for an attempt at Norwegian independence. And while we entered into a looser union with Sweden, there was a long ongoing debate about creating a new, formalised Norwegian language.
We ended up with two, but one of them - bokmål - was basically Danish + Norwegian pronunciation + various spelling reforms, as this was effectively what was spoken amongst the upper classes in the cities, and this (with further spelling reforms) is the dominant written language today (the other was based on spoken dialects, and is further from Danish but not that much) .
So we've "only" had about 200 years for the languages to diverge again.
Place with the google translate pronunciation button for some simple sentences for a while - you'll be surprised how much you understand