We experimentally investigate the problem of monitoring the Kerr-nonlinearity coefficient $\gamma$ from transmitted and received data for a single-mode fiber link of 1600 km length. We compare the accuracy and speed of three different approaches. First, a standard split-step Fourier method is used to predict the output at various $\gamma$ values, which are then compared to the measured output. Second, a recently proposed nonlinear Fourier transform (NFT)-based method, which matches solitonic eigenvalues in the transmitted and received signals for various $\gamma$ values. Third, a novel fast version of the NFT-based method, which only matches the highest few eigenvalues. Although the NFT-based methods do not scale with link length, we demonstrate that the SSFM-based method is significantly faster than the basic NFT-based method for the considered link of 1600 km, and outperforms even the faster version. However, for a simulated link of 8000 km, the fast NFT-based method is shown to be faster than the SSMF-based method, although at the cost of a small loss in accuracy.