X6mh @10K ~2 X PI X 10^4 X 6 X 10^-3 ~ 360 ohms @ 1A = 360V !! So putting in a cap of -j 360, gets this down to the series loss resistance of the coil, at 5 KHZ, Xl = 180, Xc = 720, total Z = -j520, so now you need 520V for your current unless you times the C by 4. if we look at it the other way and say you have 200V available and arrange for the circuit to look capacitive at 10 KHZ, then Z = 200 or +j360 and -j 160, so at 5KHZ, its +j180 and -j320 or Z = -j140, which is OK. So with a200V supply you can possiblely squeeze a frequency range of 10KHZ to 4 KHZ, or 2.5 :1, you need 10:1, so you could just about get away with three auto switched in capacitors.
As explained high Q components need to be tuned to the correct frequency, you have a fixed L so you need a massive size (I = 2A) variable cap with range of 100 :1 , this is not possible.
With a huge voltage supply you can get the high frequency current, in fact it seems that you are having to trade PSU voltage against bandwidth or number of switched capacitors.
Frank