The ensemble Kalman filter (EnKF) is a Monte Carlo approximation of the Kalman filter for high dimensional linear Gaussian state space models. EnKF methods have also been developed for parameter inference of static Bayesian models with a Gaussian likelihood, in a way that is analogous to likelihood tempering sequential Monte Carlo (SMC). These methods are commonly referred to as ensemble Kalman inversion (EKI). Unlike SMC, the inference from EKI is only asymptotically unbiased if the likelihood is linear Gaussian and the priors are Gaussian. However, EKI is significantly faster to run. Currently, a large limitation of EKI methods is that the covariance of the measurement error is assumed to be fully known. We develop a new method, which we call component-wise iterative ensemble Kalman inversion (CW-IEKI), that allows elements of the covariance matrix to be inferred alongside the model parameters at negligible extra cost. This novel method is compared to SMC on three different application examples: a model of nitrogen mineralisation in soil that is based on the Agricultural Production Systems Simulator (APSIM), a model predicting seagrass decline due to stress from water temperature and light, and a model predicting coral calcification rates. On all of these examples, we find that CW-IEKI has relatively similar predictive performance to SMC, albeit with greater uncertainty, and it has a significantly faster run time.