Northern is failing for much of the same reasons as GTR didn’t work properly to start with. Too large an operation with too many different routes having completely different needs. It really should be split into at least two TOCs, ideally three.
Is fragmentation a good thing though?
It could be split into operating divisions, a la South Eastern, South Central and South Western Division of the Southern Region, if it is not already (and such things still exist) but separate companies just means less integrated timetable planning and multiple TOCs competing for paths.
It seems natural to me to have a single Northern TOC, because there is a lot of cross-Pennine traffic and routes so even the old LM/ER split doesn't make a lot of sense.
I'd go the other way and merge TPE and Northern, because they are both in the same area of the country (though obviously keep the long-distance stock with the long-distance services!)
My number one beef with privatisation was the fragmented nature of it; while politically I prefer public ownership for the railways I would have been a lot happier with a single franchise, or failing that, a small number of large regional TOCs (e.g. Southern, East Anglia, Great Western, Northern* and Scottish), running all passenger services (with regulated fares), to maximise integration and minimise bureaucracy.
(*basically combined LMS and LNER, minus Scotland and East Anglia, to produce a common TOC both sides of the Pennines).