How does the key size per data bit influence the security? The likelihood of breaking, for instance, an AES-128 cipher is 100% after $2^{128}$ tries in brute force, meaning that I've got to try $2^{128}$ keys to certainly break it.

What if I (hypothetically) had recorded $2^{128}$ bits of data encrypted with the same AES-128 key? How does this help in cryptoanalysis?

And how is the situation differently if or if not a part of this data (e.g. word file headers, specific telnet handshakes, whatever) was known plaintext?
Quite apart from the correction that Reid made (it takes $2^{127}$ attempts to achieve a probability of 50% of finding the right key; with $2^{64}$ attempts, the probability of success is $2^{-64}$), with AES, there is no known way to take advantage of known (or even chosen) plaintext to speed up any brute force search; even with $2^{64}$ chosen plaintext/ciphertext pairs, the best known attack is to pick one pair, and apply a brute force search on that pair.