We don't know, but general relativity implies that no force can stop the object from shrinking to infinitesimally small. Quantum Mechanics is more uncertain.
I just want to point out that 10 solar masses would have a radius 29.53 km at the event horizon, not 1 KM. A Neutron Star is pretty close to its event horizon already.
The problem with "some distinct size inside the event horizon" is that things change when the acceleration is greater than the speed of light. If the acceleration is strong enough that the gluons are drawn towards the center faster than they an move (speed of light), then it's entirely reasonable that the structure wouldn't hold. Another way of saying this is that the gravity becomes stronger than the Neutron Degeneracy Pressure and there's no known force that would prevent continued shrinkage once the Neutron Degeneracy Pressure is overcome.
The Pauli exclusion principal says that two neutrons can't occupy the same space, but this can be overcome somewhat by the weight and pressure approaching infinity. The neutrons can get closer and closer as their momentum approaches infinity and that happens at the singularity.
So, both relativity and QM (the two best theories for dealing with this question) say that a ridiculously small object may exist at the singuarity, not meters, but maybe the size of an atomic nucleus or size of a quark or even, possibly zero size, but without a quantum field theory of gravity it's impossible to say with much confidence.
Other possibilities exist like space might get stretched into some very strange properties, where virtual particles become real but fall back towards the singularity, similar to hawking radiation but without the escape, or, the angular momentum could form into a ring or band, not a point but all that is just guesses. The mathematics of the two main theories both say that a very tiny point is entirely reasonable and the argument for what you suggest, an object with a distinct radius of say 1 meter, is far less supported by current theories.
That's my layman's answer anyway. I can delete this if I got too much of the physics wrong, but I thought I'd give you a quick answer.