The International Muon Ionisation Cooling Experiment (MICE) is currently being commissioned at the Rutherford Appleton Laboratory (RAL) and aims to be the first experiment to demonstrate the technique of muon beam ionisation cooling, a key component in Neutrino Factory and Muon Collider designs. In order to do so, high precision muon track and beam phase-space reconstruction is required. A Kalman Filter based track fit has been designed, implemented and tested for the reconstruction of both straight and helical muon tracks, using data from the MICE Scintillating Fibre Trackers. Detailed Monte Carlo simulation studies have been conducted in order to verify the implementation and predict the performance of the fitting algorithms when applied to real data. A combination of simulation studies and statistical data analyses have been used to commission the track reconstruction software and perform simple validation techniques. The straight track reconstruction was applied to an alignment study, designed to measure the relative positions of the two tracking detectors within the MICE cooling channel. A translational resolution of 0.11 mm in both the x and y directions was extracted, in addition a rotational resolution of 0.05 mrad of the relative yaw, Φ x , and pitch, Φ y , between the two trackers was calculated. The helical track reconstruction was successfully used in the analysis of the first helical track data acquired within the MICE Cooling Channel. A momentum window of 195 to 205 MeV/c was applied to the default MICE muon beam, which resulted in 5049 reconstructed tracks with a normalised transverse emittance of 5.08 ± 0.05 Stat ± 0.06 Sys ± 0.02 Field mm. The systematic error was calculated based on the modelled momentum residuals, and a second systematic error, due to the incomplete knowledge of the true field, was estimated through comparisons of different field maps.