A trillion neurons properly added to a human brain would give us very good tactical and analytical abilities, but likely wouldn't be enough to compete in real-time with advanced AI due to latency issues. However, it should give us a pretty good boost in the research and development department, which will allows us to reverse-engineer the alien technology and physiology to create supersoldiers and our own battle robots.
Google indicates only about half the world population lives in urban areas0, so after the purge, we've still got about three billion humans. That's a 6000:1 advantage over the aliens. If we can stand up production lines in several areas around the world and pump out supersoldiers before the aliens realize what we're up to, we might stand a fighting chance.
Humans are Slow
If it's just a matter of combat ability, intellect isn't really that important. It's a lot more about speed and strength. If the AI thinks on micro-to-millisecond terms, it's highly unlikely we could ever compete with them just by growing bigger brains. In fact, we might do better to grow smaller brains.
The problem is human reactions times are between 100 ms and 500+ ms for really easy tests (like, press the button when a giant horn turns on in your ear)1, and upwards of 1000 ms (a full second) to make cognitive decisions that require motor control (like, a cop decides the bad guy is attacking, then draws his gun and fires)2.
The cause of this is three-fold. First, synapses and neurons are terribly slow compared to electronics. Signals travel around a hundred meters per second in organics3, compared to a quarter of a billion meters per second in electronics4; and the switching time of a neuron is around a millisecond5, compared to nanoseconds for electronic gates (and plausibly much faster for atomic or quantum computers)6.
Second, we have really big brains and bodies, so it takes a long time for the signal to propagate all the way through our brains, then a really long time for the signal to get to our muscles. The signal goes through millions of parallel paths, so the total information processed isn't terrible, but the latency is huge.
Third, we're too smart. We don't just think in terms of "bad guy, best way to kill, execute". Instead, we think "shiny metal thing, reminds me of that movie when I was eight, probably a bad guy, I think that gun is going to kill me but it looks pretty awesome, man I hope I don't poop myself because that girl back at camp might not go out with me if I come back with poop running down my leg, I should probably run now". We can train our brains to bypass most of the extraneous crap during combat, but it's always leeching a little processing power away. Plus, having that capacity requires our brains to be larger, which leads back to the second issue.
Insects can have reaction times as low as 5-10 ms7, because their tiny, simple brains do a few things really well, instead of our brains that do a lot of things decently.
On the other hand, if we're just bigger, stronger, and outnumber them ten thousand to one, it would just be a matter of attrition. Even if they're quick enough to dodge our attacks for a little while, they'll eventually get tired. Then we destroy them. It's how we've been hunting animals for millennia, if not longer.8
Humans may be Weak
I don't know anything about your aliens, but if they're made of impenetrable unobtainium and hit with the force of Thor's hammer, speed won't really help much. If they can invent toxic gases to destroy the planet before we've cracked the first alien's exoskeleton, we're kind of screwed.
From your edit, it seems like their military tech would likely put them pretty close to this scenario, even if the aliens themselves are relatively squishy. So a head-on assault would likely be met with brutal defeat, even if we had the reaction times of a cockroach and the strength of a whale.
More neurons in the right places could potentially give us much better capacity for pattern recognition, predictive capabilities, behavioral analysis of the alien invaders, and other cool things that could help us with large-scale tactics, although we still need the capacity to actually hurt them.
One of the better uses of enhanced cognitive abilities would be reverse-engineering alien tech, or even reprogramming a bunch of the aliens to fight on our side. Once we have enough tech to punch through whatever armor they have, and withstand an onslaught from their weaponry, then we stand a fighting chance.
On the other hand, if we have the technology to turn humans into walking supercomputers, we could probably build a staggeringly powerful actual supercomputer and just let it do the analysis. Maybe plug one or two fairly typical humans into the supercomputer to nudge it in the right direction from time to time.
A more useful method might be to combine the two ideas. Build a supercomputer with a few humans permanently plugged in as interfaces and controllers. It's rather barbaric, but given the fate of our species is on the line, it's probably justifiable.
Depending on the aliens' speed and strength, we may need a combination of thinking faster and all that alien tech to beat them in face-to-face combat. In this case, we'd want to modify our soldiers' neural circuitry so all their combat algorithms are running on something similar to the alien AI circuitry. We could modify the other algorithms (or remove them entirely), but as long as they can react quickly to changes on the battlefield, they'll be competitive, and removing their human natures might have unexpected consequences.
We'll also want to build vehicles. Tanks and aircraft can do enormous amounts of damage compared to a normal human. Of course, we need alien-tech-infused tanks and aircraft. Then we'll need to build super-pilots to drive and fly them. Better yet, we could just build them straight into the vehicle to give them the best possible response times, like KITT from Knight Rider, or the Cylon Raiders from new BSG.