You have misunderstood the MIT license.
The MIT license requires you to include a copy of "the above copyright notice and this permission notice" in "all copies or substantial portions of the Software." However, this is not the same as requiring you to offer the Software under those terms, and in fact the MIT license explicitly permits you to sublicense the Software under different terms ("including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software"). If you had to offer the Software under the same terms, then the sublicense right would be extinguished, so that cannot be the correct reading of the license. Therefore, you can do exactly what you describe, releasing the software under the GPL.
In other words, you have to include the permission notice, but you are expressly permitted to wrap it in terms that state "these permissions don't apply to you, dear end user, but only to people who go and download the original from the upstream source." That's what "sublicense" means.
The Free Software Foundation, which publishes the GPL, agrees with this analysis (although they recommend avoiding the phrase "MIT License" because MIT has offered software under other licenses).