We make modifications to the unscented Kalman filter (UKF) which bestow almost complete practical identifiability upon a lumped-parameter cardiovascular model with 10 parameters and 4 output observables - a highly non-linear, stiff problem of clinical significance. The modifications overcome the challenging problems of rank deficiency when applying the UKF to parameter estimation. Rank deficiency usually means only a small subset of parameters can be estimated. Traditionally, pragmatic compromises are made, such as selecting an optimal subset of parameters for estimation and fixing non-influential parameters. Kalman filters are typically used for dynamical state tracking, to facilitate the control u at every time step. However, for the purpose of parameter estimation, this constraint no longer applies. Our modification has transformed the utility of UKF for the parameter estimation purpose, including minimally influential parameters, with excellent robustness (i.e., under severe noise corruption, challenging patho-physiology, and no prior knowledge of parameter distributions). The modified UKF algorithm is robust in recovering almost all parameters to over 98% accuracy, over 90% of the time, with a challenging target data set of 50, 10-parameter samples. We compare this to the original implementation of the UKF algorithm for parameter estimation and demonstrate a significant improvement.