Starting Issue On 1999 V plate Kawasaki Ninja ZX6R?

Basically I have recently bought the above bike. And since then it will not start.
I have tried as much as I know with and without the choke on, In gear and out of gear, and with and without the stand up and down.
I have ruled out that it is not the battery that is causing the problem, it is fully charged and will happily crank the engine all day without much labouring. So Im lead to believe it is a fuelling or sparking issue. I have taken out the spark plugs and cleaned them using carb cleaner and drying them back down with a cloth and Ive cleaned and put some WD40 on the ends of the HT leads. I still have the same issue and nothing has been resolved by this. Therefore I am now a little stuck as to what to do next. There is fuel in the tank and it WILL start from bump starting/push starting it. After this has been done it will not idle until after a couple of minutes when it has warmed up. Then I can pretty much start it when I want. The actual performance of the bike seems completely unaffected when riding and it will happily idle at junctions etc when it is warm.
I'm currently thinking that the spark plugs may need replacing but really I just want/need opinions.

Any help is appreciated. Andy
Absolute spot on answer that mate. Took seat off and took the relay off, tapped it with a driver and cleaned all the terminals with a small wire brush, used a drop of WD40 on the connectors, starts first time every time now. - However the bike is still warm so ill see how we go...
Thanks again!

change the start relay
when the starter motor is running its pulling all the voltage from the battery and the ignition circuit isn't getting enough power to the coils causing a weak spark to the spark plugs.
there is a bypass circuit ( look on a wiring diagram ) that the ign coils get full battery voltage, a direct connection from the battery only when the starter is engaged, this circuit comes from the starter relay ( not the starter solenoid )
this can be verified by jumping a wire direct from the battery to the + of the ign coils

note: if you do use the jumper wire, the engine won't shut down untill you remove the jumper wire, ( turning the key or kill switch off won't stop the engine )
but before i replace the relay I would carefully inspect the plug in wiring harness connections for corrosion or a weak electrical connection, clean with a miniature brass wire brush and apply generous amounts of di-electric grease to the connectors. I have seen the wiring harness connectors corrosion problems in many bikes not even as old,
I stress the importance of using di-electrical in the wiring harness connectors of all bikes and atv's