Amplitude Estimating Using a Windowed DFT
I have several blog articles that solve this problem exactly in a theoretical sense, and quite accurately in an implementattion.

3. DFT Pure Tone Frequency Formulas


Exact Frequency Formula for a Pure Real Tone in a DFT
Two Bin Exact Frequency Formulas for a Pure Real Tone in a DFT
Improved Three Bin Exact Frequency Formula for a Pure Real Tone in a DFT
A Two Bin Exact Frequency Formula for a Pure Complex Tone in a DFT
Three Bin Exact Frequency Formulas for a Pure Complex Tone in a DFT


4. DFT Phase and Amplitude


Phase and Amplitude Calculation for a Pure Real Tone in a DFT: Method 1
Phase and Amplitude Calculation for a Pure Complex Tone in a DFT
Phase and Amplitude Calculation for a Pure Complex Tone in a DFT using Multiple Bins


The rest of the articles are pretty good too. ;-)

I posted one this morning:


Angle Addition Formulas from Euler's Formula