Characterizing enzyme kinetics is critical to understand cellular systems and to utilize enzymes in industry. To estimate enzyme kinetics from reaction progress curves of substrates, the Michaelis-Menten equation has been widely used for a century. However, this canonical approach works in a limited condition such as a large excess of substrate over enzyme. Even when such condition is satisfied, identifiability of parameters is not guaranteed, and criteria for the identifiability is often not easy to be tested. To overcome these limits of the canonical approach, here we propose a Bayesian inference based on an equation derived with the total quasi-steady state approximation. Estimates obtained with this approach have a little bias for any combination of enzyme and substrate concentrations in contrast to the canonical approach. Furthermore, with our new approach, an optimal experiment leading to maximal increase of the identifiability can be easily designed by simply analyzing scatter plots of estimates. Indeed, with this optimal protocol, kinetics of diverse enzymes with disparate catalytic efficiencies such as chymotrypsin, fumarase and urease can be accurately estimated from a minimal number of experimental data. A Bayesian inference computational package that performs such accurate and efficient enzyme kinetics is provided in this work.