With only on/off you can likely omit the drivers just use a resistor to gate and use logic level or better gate threshold MOSFETs... I agree with Goeytex - do be careful to not exceed gate voltages specified in data sheet...
I think you need to give us more details of the LEDs. Can you give a part/model number or a URL to the LED or data sheet? Goeytex has drawn each of them as a simple, single LED in the diagram. They will definitely not be that simple at 10 watts each!
10 watt LEDs come in many varieties and some can require nearly 1 Amp. If they are a bare 10W LED emitter, you need to consider current limiting otherwise they will quickly overheat and destroy themselves.
IWP is correct. There likely needs to be current limiting of some kind. How this is implemented depends upon the 10 Watt LED Arrays.
My assumption was that since the OP had previous success with 6 watt units, that the need for current limiting was understood. I will add a current limit block to the diagram to eliminate any confusion.
Warning:Automotive electrical systems can generate transients > 60V. In automotive applications it is good design practice to use automotive grade regulator(s)/ components and to provide transient suppression devices such as TVS Diodes to protect the Picaxe and other components from these transient voltage spikes. Failure to do so could lead to failed components.
Failure to provide proper current limiting to the LEDs will result in overheating and failure of the LEDs. The previously attached circuit will not provide a constant current to the LED's over the range of the automobiles +12V supply. (12v - 13.5v)
A much better solution would be to use a constant current source or a commercial LED controller for driving the LEDs.