It will be VERY hard to find a real wood floor (engineered or solid) in a fashionable gray for less than $5-$7/sf. Even then the fashionable colours are always 20% more than the "ugly" floors. You can one of several things:

1) Raise your flooring budget allow $7/sf...this will give you a fighting chance

2) Look at vinyl plank flooring = $4.50/sf is a very nice vinyl = appropriate for kitchens

3) Look at "wood look tiles" - This will increase your INSTALLATION costs...but material costs will be nicely inside your budget (should be more expensive than a wood floor)

4) Sheet Vinyl in a wood-look pattern. Material costs might be a bit higher but install should be a snick lower ... it will work out in the wash.

Those are your options. Do NOT use laminate in a kitchen.