Could an audio induction loop be used to transmit wi-fi?
Possible, yes. Useful, no.

An audio induction loop is a loop of wire around a room or building that is driven with baseband audio frequencies. Assuming we are talking about audio for humans, these frequencies are in the range of 20Hz to 20000Hz. Thus, the highest audible frequencies have a wavelength of about:

$$ \lambda = \frac{c}{f} = \frac{300 \cdot 10^6\:\mathrm{m/s}}{20000\:\mathrm{Hz}} = 15\:\mathrm{km}$$

It's unlikely any practical loop around a building will be anywhere near this large, so we can consider an audio induction loop to be a small loop antenna. This means that the time it takes an electric wave to travel around the loop is very small compared to the wavelength, so we can consider the current in the loop to be the same everywhere. Essentially, the audio induction loop is one half of an air-core transformer, with the other half being the receiving device, which must be in the near field of the loop to function effectively.

On the other hand, of the many Wi-Fi standards, I don't believe any common standard operates at a frequency less than \$2.4GHz\$. The wavelength is approximately:

$$ \frac{300 \cdot 10^6 \:\mathrm{m/s}}{2.4\:\mathrm{GHz}} = 125\:\mathrm{mm} $$

This is orders of magnitude smaller than any imaginable audio induction loop. Thus, the current in the loop wouldn't be the same everywhere, because in the time of one cycle, the electric wave has propagated only a small distance around the loop.

This complicates the analysis of this antenna significantly. Analysis is especially complicated in the near field, because you will get a pattern of standing waves that work constructively in some locations, and destructively in others. The ordinary objects in a room will also interact with the antenna, refracting, reflecting, and absorbing the electric energy in the room, making any simple analysis not practical.

That's not to say it couldn't be done, but given that every Wi-Fi access point already has an antenna that does work, what would be the advantage?