Inverse problems in cardiovascular modelling have become increasingly important to assess each patient individually. These problems entail estimation of patient-specific model parameters from uncertain measurements acquired in the clinic. In recent years, the method of data-assimilation, especially the unscented Kalman filter, has gained popularity to address computational efficiency and uncertainty consideration in such problems. This work highlights and presents solutions to several challenges of this method pertinent to models of cardiovascular haemodynamics. These include methods to a) avoid ill-conditioning of covariance matrix; b) handle a variety of measurement types; c) include a variety of prior knowledge in the method; and d) incorporate measurements acquired at different heart-rates, a common situation in the clinic where patient-state differs between various clinical acquisitions. Results are presented for two patient-specific cases of congenital heart disease. To illustrate and validate data-assimilation with measurements at different heart-rates, results are presented on synthetic data-set and on a patient-specific case with heart valve regurgitation. It is shown that the new method significantly improves the agreement between model predictions and measurements. The developed methods can be readily applied to other pathophysiologies and extended to dynamical systems which exhibit different responses under different sets of known parameters or different sets of inputs (such as forcing/excitation frequencies).