Impossible to know without full information about the performance of the core. Chances are the "Iron Losses" will go through the roof at that frequency, and it will rapidly overheat. Try doing some tests with an audio signal generator, a dummy load and an oscilloscope.
If we know the full information about the toroidal core, is it possible to calculate all the parameters in Model A and Model B? Thanks.
No, to find all those parameters you need to measure them from the actual transformer. They are determined by far more than just the core. Also, model A (The 'T' model) and B (The 'Cantilever' model) are the same, just with different representations. The 'T' model is more common.
Also, there are
many different types of ferrite, and each is designed to operate in a specific band of frequencies. Powdered iron is something completely different and is generally used for inductors (chokes) rather than transformers, when energy storage is required. 10kHz would be OK for laminated iron (at the top end though), but ferrite would be much lower loss, and probably more appropriate.
Since you're using the transformer well under the maximum power rating, it'll probably work, but it'll also probably be rubbish (i.e. lossy, and with poor regulation).
HOWEVER... bear in mind that you seem to be using the transformer backwards in your appplication. Something for which it was not designed at all. If there are not enough turns on the secondary winding (your primary) then you may saturate the core, potentially destroying it and damaging the circuits connected to it.
To figure out if you're going to saturate it, you need to use the calculation:
Bmax= Vrms/(4.44*f*N*A)
This is only valid for sinusoidal signals, but it's close for others. If it's an iron core, you need to keep Bmax under about 1T, and if it's ferrite then 200uT is a good rule of thumb. You need to know N (the number of turns) and A (core area) which might be tough if the datasheet isn't really good.