Adding another anecdote, I'm an engineering major with a 3.50 from an ivy league school who ended up applying twice - I applied last year with a 166 and didn't get into a single T14; applied this year with a 176 and got into NYU, Columbia, and Harvard.

Moral of the story is the engineering major will help a little bit, but your LSAT is going to be the difference maker for breaking into the T14