It was discussed recently by Ivan Portillo in a very well-written article that Kansas State will never be a rival to Kansas like Missouri was. It's true, the Border War featured conflict from areas of history way beyond sports, and the sports history was nothing to sneeze at in itself. But that rivalry is, at least on the field, dead. Mizzou moved on to the SEC while KU stayed in the Big 12. The Jayhawks' strongest rivalry was effectively ended at that point (in terms of actual athletic competition).
Mizzou was not the only school to leave the conference however (although they did leave the biggest impact on the Jayhawks), as Texas A&M also left for the SEC, Nebraska (another longtime rival of Kansas) left for the Big Ten and Colorado moved on to the Pac-12. The Big 12 was not the only conference to get rocked by realignment, but it was affected to the point of obvious instability. Even with the addition of TCU and West Virginia, the shaking up of the Big 12 is not over. To me, I see two potential courses of action for the teams in the Big 12 as realignment of conferences inevitably continues: more teams are added to bolster the conference, or it dissolves and the Jayhawks will be looking for a new conference.
If the Big 12 does survive, more teams will likely have to be added. Is this a good thing? Not necessarily. I, for one, am a big fan of the round robin football scheduling and I do not really mind not having a conference championship game, because the way things are shows true superiority in conference. Nonetheless, you would have to expect that more teams will be added in order to keep the Big 12 afloat (and the conference would want the added revenue from a championship game). Arkansas would be a great addition in a perfect world, but the reality is that they would never leave the SEC (no team would).
Realistically, Central Florida seems like a good team to add. They have a booming football program (the second best in the state of Florida last year after national champion ) and would benefit from a larger conference with more exposure and better TV contracts. Plus, this would expand Big 12 recruiting territory even further (after the addition of West Virginia expanded it originally). UCF definitely helps Big 12 football prestige, although in basketball is does not add nearly as much. Even so, the added benefits in football is of more importance to the Big 12 at this point and UCF would be a great team to add.
Of course, you cannot stop with eleven teams, because you still need one more to get the championship game and even divisions. Unfortunately, many teams are already in good positions within good conferences so the likelihood of adding another good or great team is relatively slim. Still, there are a few that could be poached away (although they are all teams that are in non-P5 conferences for the most part). While Texas certainly has a few candidates (SMU and Houston come to mind), they do not expand recruiting territory or anything like that. If the Big 12 was looking to take a huge gamble, extending an offer to Northern Illinois would expand the conference's reach into the Chicagoland area. Unfortunately for NIU, that invitation is likely not coming as they are still unproven against the better teams in FBS, even with their success in the MAC. That's a huge and highly unlikely gamble.
Personally, I feel as though Cincinnati would be a solid choice to be a twelfth team. The Big 12 gets a presence in Ohio and adds a decent team in both football and basketball. It does not hurt the Bearcats to move to a bigger and better conference than the AAC, especially considering their rivalry with Louisville is dormant now since Louisville moved to the ACC. While these two potential additions would really damage the AAC, it would definitely help the Big 12 and would be their best options. In terms of rivalries within the conference, nothing changes all that much. The biggest rivalry for Kansas remains K-State, and neither team entering the Big 12 would have an existing rivalry that would heat up (although they would be leaving some behind in the AAC).
There is also the chance that the Big 12 does not add any new teams, and while the conference is fine the way it is, in this modern era of super conferences it is hard to survive with just 10 teams and no conference championship game in football. For the sake of discussion, it is entirely possible that the Big 12 does in fact dissolve. If that's the case, the ten current teams will be looking for new homes. Where would the Jayhawks end up? It would most likely be the Big Ten. Really, it's the only decent choice. For basketball, it would certainly be a much stronger conference with the Jayhawks in the mix, while for football the team would realistically be in a similar situation to what it is in right now. With all of the exposure thanks to various TV contracts (including the Big Ten Network), an even better presence in the Great Lakes region and a team that may actually finish below them in football standings (Purdue football is really, really bad), this is definitely positive for Kansas. (Assuming everything academically works out,) Kansas would benefit from joining the Big Ten and vice versa.
Rivalry-wise, joining the Big Ten is certainly beneficial for the Jayhawks. It does not seem like KU will be playing Mizzou any time soon, unfortunately, but even so there are plenty of teams that Kansas may have a legitimate, full-fledged rivalry with (which benefits the team, the league and the fans, because the best driving force in sports is rivalry). The feud with Nebraska can be reignited for the Jayhawks, something that I for one would love to see (even if the football teams are not at all on the same level right now). You add to that the fact that other teams from the Big 12 would be moving into the Big Ten along with KU, specifically in theory Iowa State and K-State. The rivalry with both would be strengthened because of the Big 12 tradition.
Also, KU/ISU has become a very underrated rivalry in recent years (at least in basketball), one that is worth watching for years to come. If nothing else, Kansas would have a new rivalry with Illinois, as Illini fans are still bitter about Bill Self. All in all, if the Big 12 were to fall apart, joing the Big Ten would be the best possible scenario for the Jayhawks (especially if they were able to retain rivalry with former conference-mates from the Big 12 like Iowa State and K-State).
The NCAA landscape is shaped by super conferences nowadays, and the Big 12 is going to need to either adapt or it will fall apart. Moving forward, conferences will continue to realign and the Big 12 will be at the epicenter of these seismic shifts. If they make some moves to add teams back to the conference, it will survive, although for Kansas specifically not a whole lot changes, as they'll still be one of, if not the, weakest football team and the strongest basketball team. If the Jayhawks are looking for a new conference some day soon, the Big Ten is their best option. (I must admit some bias; I have thought about KU joining a future 20-team Big Ten for a while now and have convinced myself it is a great idea.)
Changing conferences and whatnot is not as simple as just saying "leave here, move here, etc." as things like TV contracts, money in general, politics and more get in the way. For now we are left to ponder what if's, but one certain thing is that the Big 12 is in for more major changes in the coming years.