Hi, this can be a difficult question to approach because the Danelaw doesn't leave us any contemporary written records and DNA evidence can't really distinguish between Anglo-Saxons and Vikings; although approximately 40% of the population in the modern regions of the Danelaw are of Germanic descent, opposed to ~30% in Central England and ~20% in Cornwall, this might be the result of denser original Anglo-Saxon settlement as much as it could have been the result of Viking settlement.

I touched somewhat on your topic here, the main conclusion of which is that in literature, the Anglo-Saxons themselves took great pains to distance themselves from the Vikings as much as possible. Although it does refer to 'The Danes', the Anglo-Saxon Chronicle doesn't seem to care too much where the Vikings actually came from unless the specific origin of an army can be attested. Often they are simply 'the Pagans'. Asser, in his Vita Ælfredi, simply calls them 'the Heathens' or 'the Pagans', who stand in stark contrast to the Christian English.

Whether this tendency towards as much 'other-ing' as possible of the Vikings is a phenomenon that actually occured on the ground in the Danelaw is very hard to say. Certainly to the writers of the Anglo-Saxon Chronicle, the distinction remained, and they refer explicitly to 'the Englishmen living under the Danes'. Of course, this national definition also serves as a useful justification for the Wessex-led 'liberation' of what had been separate Anglo-Saxon kingdoms.

The Vikings themselves don't leave us any written records of the Danelaw, but archaeology suggests that they were quite keen to integrate themselves into the local English population. Numismatic evidence shows that in the 880s, they were producing facsimilies of Alfred's London-issued coinage, likely to try and facilitate trade across the border. By the early 900s, Viking East Anglia was producing coinage celebrating the cult of 'St Edmund the Martyr', who had been killed fighting the Viking invasion of East Anglia in 870.

In order to more specifically answer your questions in relation to the late tenth and early eleventh centuries, one of the most niche yet informative sources is in personal decorative metalwork, specifically strap ends. These are little metal clasps which, as the name suggests, are used to adorn the end of cloth or metal straps. These are surprisingly ubiquitous in a period where belt buckles were out of fashion and clothes rarely had fastenings. Strap-ends adorned belts, bag straps, sword belts, slings, horse trappings etc, and were often richly patterned or engraved as an outward display of fashion and wealth. For those who couldn't afford fancy gilded or inlaid belts, cheaper metal copies were usuallt produced, but still often with ornate designs for a very visual culture. There are many distinguishable English styles of design, but from the mid-tenth century, versions start appearing from inside the Danelaw with characteristic Scandinavian 'Borre' style designs, indicating that, as the Vikings were trying to integrate into English culture, so too were the English adopting Scandinavian cultural elements.

In the late tenth century, there is an indication that there was a lasting integration in the Danelaw that distinguished it from the rest of England, although again this could be due more to Northumbria's history as an independent kingdom rather than necessarily because of the Viking influence. Northumbria remains politically troublesome and periodically rebellious throughout the 10th Century until Edgar grants it a measure of relative autonomy in return for its loyalty in the 970s. He also punishes the city of Rocester after the townspeople assault merchants from the Danelaw, which indicates that even after a decades-long period of peace and supposed unity, the southern English Anglo-Saxons still distinguished between themselves and their Danelaw neighbours, apparently along ethnic lines. It also illustrates Edgar's strident political measures he followed to maintain the fragile political unity of his kingdom.