EL stuff is basically a phosphor sandwiched in between two electrodes. It electrically behaves like a capacitor with a huge dialectric loss. The amount of current flowing is dependant on the frequency as well as the voltage, dose the datasheet specify the capacitance?
At 4kHz you're better off using a ferrite core for the transformer as laminated steel will have too high an eddy loss.
Here's an inverter circuit, I've used to power flopurescent tubes.
L1 is a about 20 turns round a small ferrite core, it mearly stops surges when Tr1 and Tr2 are on simultaniously (which happens at part of the oscillator cycle).
T1's primary is 10 or so turns of 0.5mm wire wound bifilar style, the secondary is 100 turns 0.15mm. You'll probably want to adjust this ratio to give you the voltgae you want, you'll probably want to use more tuns on the primary and secondary to lower the frequency.
C1 and C1 might be required to get the oscillator running, it it works without them then good for you. I'd start off with 1nF then adjust their value to give best results.
R1 and R2 depend on the gain for Tr1 and Tr2, and will also need adjusting, use between 1k and 10k, start with 4k7.
Tr1 and Tr2 can be any general purpose transistor, but check the power ratings first.