We consider simultaneous wireless information and power transfer (SWIPT) in MIMO Broadcast networks where one energy harvesting (EH) user and one information decoding (ID) user share the same time and frequency resource. In contrast to previous SWIPT systems based on the information rate, this paper addresses the problem in terms of the weighted minimum mean squared error (WMMSE) criterion. First, we formulate the WMMSE-SWIPT problem which minimizes the weighted sum-MSE of the message signal arrived at the ID user, while satisfying the requirement on the energy that can be harvested from the signal at the EH user. Then, we propose the optimal precoder structure of the problem and identify the best possible MSE-energy tradeoff region through the alternative update of the linear precoder at the transmitter with the linear receiver at the ID user. From the derived solution, several interesting observations are made compared to the conventional SWIPT designs.