Or, you could gather your quotes for the A/C, then get the money from your tenant in cash. Once the A/C is installed, pay the note, and take the capital improvement deduction over time as the tax code demands.
But, in all seriousness, I would do what the previous post says: Your tenant knew what they were getting in, so tell her NO, and move on. Make her adhere to the lease agreement, then when the lease terms are up, give her 30 days notice to move out.
Your tenant seems like a flake. Besides, I wouldn’t be surprised at all to find that your tenant has not a clue about how much a central A/C system costs.