Estimate amplitude of signal between 45 Hz-60 Hz
What you describe is a very simple signal model: one tone in a constrained frequency range, and superimposed non-sinusoidal noise. 

You can do a lot of things based on that. For example, you could do the FM radio / lock-in amplifier dance, and just use a PLL (with a low control loop filter bandwidth, to remove most of the noise!) to lock onto the tone; using the resulting tone to mix your signal, aggressively low-pass filter it and simply take the average amplitude of the resulting signal.

You could also use (other) parametric estimators. 

So, to me, estimation would roughly work like this:


band pass filter so that you only retain the signal of interest; preserving a bit of roll-off for the filter, that'd be something like filtering to 40–65 Hz. That gives you a remaining information-containing signal bandwidth of 25 Hz, so decimate your signal in the filtering step down to a sampling rate of 50 Hz. The rest of the signal doesn't contain anything you care about, so you're not losing anything by doing so!
The resulting signal will only contain the little noise power that falls into that bandwidth, plus the signal we care about. So, that's really a simple signal model now! Apply a suitable parametric estimator. 


W.r.t. what a suitable parametric estimator would be:


If your SNR in this very reduced bandwidth is very good (and that's not unlikely!), calculate the analytic signal, and simply calculate the arctan of the ratio of successive samples of that. Apply some averaging/low-pass filtering to the result to get rid of remaining noise influences. It's not fast, it's not overly stable, but it's easy to grok (frequency is the angular velocity with which the signal runs along a origin circle in the imaginary plane). Frequency resolution: Defined by the accuracy of your arctan implementation (and of course limited by noise).
If your SNR is so-and-so, and you can live with limited resolution: Take $m$ samples of input. Concatenate $N-m$ zeros, so that you get an $N$-long zero-padded vector. Do the FFT of that. You'll get a PSD estimate with $N$ equidistant points within your bandwidth. Find the three strongest adjacent bins. You know the actual frequency of your tone is somewhere between these, on the maximum of a sinc running through these three bins. Since close to its maximum, a sinc is pretty similar to a quadratic function, use the highschool formulas to find the maximum of that sinc from the three points. Resolution is mainly limited by SNR, but this algorithm is somewhat more sensitive to noise than others.
Else, I'd recommend doing ESPRIT. A bit over the top, but gives you a frequency estimator that's pretty close to the Cramér-Rao Lower Boundary for short observations, and pretty stable in the presence of non-sinusoidal noise. Also, easy to adapt to more than one tone (simply change one parameter). The calculations involved (one vector-vector product, one quasi-pseudoinverse, two eigenvalue decompositions) are benign enough for your rates.