The Linn Classik K CD/Receiver is actually a very impressive piece of hi-fi gear, in spite of its diminutive size and "jack-of-all-trades" goal.
I have owned much gear over the past two decades ranging from the heavy duty American amp's made by Krell and Audio Research, to British electronics from Quad, Naim Audio, Leak, etc.
I do have to say that the Brit's have something going for them in that most of the more charismatic companies (including the aforementioned) do manufacture equipment that faithfully honors the music -- even their lower priced gear.
You did not mention whether or not you wanted to upgrade with new or used components.
If I many offer a suggestion here. Given that the used market is literally flooded with excellent used audio equipment, it would seem foolish to spend your hard earned dollars paying a premium for new gear; gear which may not even offer a substantial improvement in sound quality over the older equipment which it has replaced.
For example, Naim Audio components (perhaps the most respected name in British Audio) built during the '90's offer an excellent opportunity to the budget conscious audiophile, to improve their present setup.
Since most of Naim's hi-end gear was prohibitively expensive at the time, most of us could not have afforded to purchase it new.
However, more than a decade later, Naim gear from the '90's offers superb sound at far more reasonable prices.
Having owned much Naim gear over the years, I would suggest looking here first, since if you like the sound of the Linn but want a more detailed and controlled listening experience, the Naim components are definitely worth a listen.
I would suggest anything from a 72/hicap/140 on up, since this amplification will give you roughly the same power that you had with the Classik but with a noticably better outboard power supply and improved sound stage.
If you are able to spend a lot more, then perhaps the equipment which was manufactuered towards the end of the
decade might be more to your liking.
Truthfully, one of the sweetest sounding setups that I ever had in my diminutive listening room was a 72/Hicap/135 setup
fed by a Naim CDI and run into a pair of ancient but still
wonderful sounding Rogers LS3/5A's (15 ohm version in teak that I bought new in 1986).
While some might question the logic in running a pair of 135's into a pair LS3/5A'S I can tell you that the improvement even over an NAP 250 was noticeable. I could listen for hours without suffering any fatigue, and always looked forward to my next listening session.
Anyway, best of luck in your search for the hi-fi holy grail. While I never did completely find it in over 20 years of searching, I have had a heck of a good time trying to, and discovered music along the way.