A useful rule of thumb for bath vanities is to use a minimum of two 60-watt incandescent bulbs, which typically produce approximately 800 lumens each for a total of 1,600 lumens. In your application you have several choices:
You could choose a fixture with standard medium-base sockets and install LED bulbs that fit the sockets. You would need LED bulbs equivalent to the standard 60-watt incandescent bulbs with approximately 800 lumens each.
Considering there is only one fixture in your bathroom, you could go to a higher lumen output, and perhaps go to a 75-watt equivalent and use a dimmer, which would allow you to lower or increase the light level as needed.
You could use an integrated LED fixture, which does not have light bulbs but instead is integrated with the LED lighting as a permanent part of the fixture. LEDs have extremely long life (about 50 times that of incandescent) making this essentially a lifetime fixture purchase. With this option you would want a fixture in the range of 1200-2000 lumens. Integrated LED fixtures are more efficient in delivering light, making it possible to go with a slightly lower lumen output (1200 lumens).
To look at the many options available, visit an ALA-member showroom where a certified lighting consultant can show you a variety of design options.