AM radio antenna optimization
Generally we do not think of antennas as transmission lines, but rather loads. In this case, the problem is designing a matching network (LC tank) to match your antenna to the diode/envelope detector (receiver). In this case, matching means you transform the antenna's impedance to the complex conjugate of the receiver's impedance (or vice versa).
The tool you would want to use for this is a Smith Chart. I would use ADS's smith chart tool. I think you can get a free trial if you don't have access.

Treat the antenna as an impedance Z = R + jX. R is a combination of the radiation resistance of the antenna and ohmic loss. In your case, X may be negative, meaning that the antenna will be capacitive.

The impedance Z of the antenna is calculated just like any other network (just a bit more arcane). Practically, you can measure the complex reflection coefficient (S11) using a network analyzer and back-calculate Z.


$$ Z = \frac{1+s_{11} }{1-s_{11}}$$
If you don't have a network analyzer, some simple antennas will have analytic formulas (short dipole example).

Adding a resistor is not the worst idea, but it could dramatically decrease the efficiency of the network. In this case, you would be looking at the parameter S21 of the matching network when terminated by the appropriate loads. It may offset the power gain you get from matching.

Your analysis is correct, the bandwidth of the overall network would be very small (this is on purpose!) However, your figure contains the solution: changing the value of the variable capacitor will change the resonant frequency of the LC tank, which will change the frequency of the perfect match, which will change the frequency of the radio station.