There are no phenomena that are only perceptible through internal experience. (A term that you've basically introduced in a True Scotsman fashion.)
Consider a radio antenna. A radio operates by responding to electrons sloshing around in a wire. The radio can 'experience external reality' iff that sloshing behaves analogously to something 'external': another antenna. This kind of analogy is what observation is, and it basically defines what an experience is. (Though in general, it doesn't need to be external: you can easily observe internal state or have a feedback loop.) Two antenna can be said to pick up the same signal only if their responses strongly correlate.
A brain doesn't work any differently, it's just a far more complex antenna that integrates more complex signals from more diverse sources. The only thing you have to do is establish a strong enough correlation (depending on the accuracy you care to assert) and to do that, you need to pick a number of aspects of the system, measure them, and ensure they correlate sufficiently nicely.
Again, this has nothing to do with the brain per se: all scientific measurements operate on the same idea. A large enough number of correlated observations is sufficient to establish whether two phenomena are the same. Humans are so good at doing this implicitly that we don't even question if other people have emotions, thoughts, ideas, or experiences that differ significantly from our own. In fact, human development involves a great deal of social mimikry and exploration which serve to constrain people's behavior to those things which communicate shared experiences particularly strongly. (Conversely, human behavior is not so unpredictable that we can't build an understand of it.)