Yes, you can release your code under the GPL.
The MIT license is compatible with the GPL: you can combine MIT-licensed code with GPL'ed code, as long as the result is licensed under the GPL. This is possible because the MIT license explicitly allows you to sublicense the code (i.e. republish with a completely different license), and only requires you to keep the copyright and license notice intact.
Permissive licences such as the MIT license or Apache v2 license are often used for code that is intended to be available as widely as possible, with as few restrictions as possible for direct users (i.e. other programmers). Such code can be incorporated in proprietary and copyleft projects.