Everytime you perform SN2 on a tertiary carbon, your orgo book cries a little
.
Sorry, couldn't resist saying that. Anyway, like above, I would not recommend drawing an SN2 substitution on a tertiary carbon.
There are a few other routes you could take...
You could make an argument for an SN1 substitution. If you pull off the same hydrogen from the E2, you can do a methyl shift, and get a secondary carbocation (the bromine carbon).
Also, you have another elimination and substitution route. Your methyl group dangling off there. You can eliminate from there and form the hoffman product. Or, you can do SN1, and have a methyl shift, and get a 7-membered ring.
However, I'd think all of those will still be minor compared to the E2 mechanism you've drawn. I would think that would be the major product.
EDIT: Also, thank you very much for posting your attempt!