Knowing accurate joint positions is crucial for safe and precise control of laparoscopic surgical robots, especially for the automation of surgical sub-tasks. These robots have often been designed with cable-driven arms and tools because cables allow for larger motors to be placed at the base of the robot, further from the operating area where space is at a premium. However, by connecting the joint to its motor with a cable, any stretch in the cable can lead to errors in kinematic estimation from encoders at the motor, which can result in difficulties for accurate control of the surgical tool. In this work, we propose an efficient data-driven calibration of positioning joints of such robots, in this case the RAVEN-II surgical robotics research platform. While the calibration takes only 8-21 minutes, the accuracy of the calibrated joints remains high during a 6-hour heavily loaded operation, suggesting desirable feasibility in real practice. The calibration models take original robot states as input and are trained using zig-zag trajectories within a desired sparsity, requiring no additional sensors after training. Compared to fixed offset compensation, the Deep Neural Network calibration model can further reduce 76 percent of error and achieve accuracy of 0.104 deg, 0.120 deg, and 0.118 mm in joints 1, 2, and 3, respectively. In contrast to end-to-end models, experiments suggest that the DNN model achieves better accuracy and faster convergence when outputting the error to correct original inaccurate joint positions. Furthermore, a linear regression model is shown to have 160 times faster inference speed than DNN models for application within the 1000 Hz servo control loop, with slightly compromised accuracy.