It might not be impossible, but it's a long ways off, perhaps even by 26th century standards. Klingon space is vast, likely nearly as vast as the Federation and there is no reason to assume that it's any less populated. Of all those races that originated in what is now Klingon controlled space, how many of them were asked to join the Empire and how many were conquered, pillaged, and subjugated in order to serve the Empire.
It's likely most of them were conquered because let's face it...the Klingons weren't out there spreading peace and prosperity.
So while it's one thing to be allies with the Klingons in order to resolve the centuries of conflict between the Empire and the Federation and stand with them in a fight because they represent a prominent political and military power in the galaxy, it's quite another to acknowledge them as equals within the Federation by extending full membership. Not without first bringing about some representation and restitution to those worlds that have been conquered. Though realistically, by even 24th century standards, they've likely been in the system for so long that many wouldn't know how to operate on their own. And depending on the situation, the Empire might not be willing to even give them up.
The Klingons are like Hannibal Lector. You might use him to help solve a case, but you're never going to invite him into your home for dinner, because you'll be dinner. Likewise, the Klingons might stand beside you in a fight because it's the right thing to do or worth the greater good, but you don't want to really be thier friends or want them dating your daughter because their blood thirsty killers who despite all their talk about honor will stab you in the back if it suits their purpose.
This is an empire built on the backs of other worlds after all.
That being said, one way to do it, might be to do something along the lines of what state Cardassia is in post Dominion war. Cardassia is in the perfect position to become a federation member in the near future due to the complete devastation of their homeworld. They will be relying on the Federation and other neighboring worlds for decades to come. Through this joint cooperation, the Federation will be in a position to build Cardassia up as a close ally with the potential of becoming a full member, though it will no doubt take quite some time.
So if like the Cardassians, the Klingons suffered a defeat that so irrevocably destabilized their government and military that pretty much anyone who walked into the capital and claimed the job could be appointed leader, then the Federation might step in to help rebuild and in doing so, begin to build the bridges and ties needed to bring them into the fold.
It's hard to imagine though, the Klingons suffering such a defeat without the Federation either being involved as an ally and being equally damaged or being an opponent. And if its the latter, than any help the Federation might give, could be seen as an attempt to take over.