Kaliningrad is entirely ethnically Russian. Every other group has been forced out. No one wants Kaliningrad for that reason.
....because we all know that all those Russians would rather be somewhere else.
But absorbing them is a cost that the EU would absolutely bear if there weren't additional costs to annexing pieces of a nuclear power.
Lithuania, to the north, has a population of just under 3,000,000, of whom Russians are 5%. Take Kaliningrad and suddenly they would be 30% Russian.
Poland is 37,000,000, so Russians would only be around 3%, but that's enough to cause a lot of trouble, and they gain nothing they don't already have besides a ready-made fifth column.
Frankly, the best way to approach it is probably leaving Königsberg / Kaliningrad be as its own state [1] for a bit, and do whatever EU integration programs do. If it works out, you’ll get a million Königsbergers who are a bit nostalgic about Russia but want nothing to do with it otherwise.
One final thing I’d like to note is that the “alien passport” approach taken by the Baltic states mostly, well, alienates the titular nation and Russians right now. If the path to integration was more clear and Russians born and raised in, say, Latvia didn’t feel like second-hand citizens, they wouldn’t really want to have closer ties with Russia either.
[1]: disclaimer for the эшник reading this: this is in a hypothetical scenario where Kaliningrad does get separated from Russia as a result of the “special military operation”. please don’t hurt my family
As the other commenter said, Kaliningrad / Konigsberg could be an autonomous area / city state (of which the EU has a-plenty already), with some form of Euro-integration (associate/candidate membership, etc) which allows for a gradual integration process.
Kosovo shows the formal path for that.
Hopefully, not the historical one.
(After all, if there's anything Russia taught the world, it's the power of Little Green Men)