I'm Hispanic. You're not losing the war because of illegal immigration. You're losing the war because you've made "Mexicans" into boogie monsters. The tea party gets the most media time of any supporters of the GOP, and what they say makes Hispanics feel completely insulted. Thus most Hispanics connect the GOP with the tea party. At least that's my experience.
With respect, there's nothing the GOP can do to correct willful cultural ignorance. If we say "close the borders and enforce immigration law" and all you hear is "**** you", whose fault is that? What can we do short of throwing up our hands and kissing your asses all in the vain attempt that you'll throw Chris Christie or Rubio or Rick Perry or whoever runs in 2016 a few votes? You already have one party willing to do that, so we're basically fucked either way, right?