Steerable needles are a promising technology for delivering targeted therapies in the body in a minimally-invasive fashion, as they can curve around anatomical obstacles and hone in on anatomical targets. In order to accurately steer them, controllers must have full knowledge of the needle tip's orientation. However, current sensors either do not provide full orientation information or interfere with the needle's ability to deliver therapy. Further, torsional dynamics can vary and depend on many parameters making steerable needles difficult to accurately model, limiting the effectiveness of traditional observer methods. To overcome these limitations, we propose a model-free, learned-method that leverages LSTM neural networks to estimate the needle tip's orientation online. We validate our method by integrating it into a sliding-mode controller and steering the needle to targets in gelatin and ex vivo ovine brain tissue. We compare our method's performance against an Extended Kalman Filter, a model-based observer, achieving significantly lower targeting errors.