What can I do about this gap between my driveway and the street?
The gap exists for drainage.  Technically, it's a gutter.
In order to create a level surface to drive on, you need to fill at least part of the gutter - but you need to do so without impacting the gutter's drainage ability.
Any permanent change you make WILL need to be permitted by the city/county, or they will likely make you take it back out again at your expense.
A not so permanent solution, however, would be to assemble a ramp structure which can be removed and which can support the weight of your car without impeding water flow.  Something with a very rough design like this:

For the strength to surface area needed you're probably going to need to use metal.  Unless you're a welder/metal worker this may not be a DIY project at all.