A B737-800 pilot reported unreliable airspeed indications due to icing during initial approach. Flight crew went around and landed safely.

Narrative

On approach to Runway XXL; descending into clouds around 6000' with reported rime icing; we began to encounter airspeed related anomalies. Specifically; as we configured with flaps; the max maneuver/min maneuver speed bands converged with corresponding max/stall speed bars also converging on both Primary Flight Displays (PFDs). No 'IAS DISAGREE' msg; however I was unable to engage VNAV when cleared and as the bands converged it became apparent that we had a problem. I disconnected autopilot and around 2600' initiated a soft go around. As I was concerned with our stall margin; I did not change configuration until I was safely in climb with speed increasing. Our airspeeds were in sync with each other and the groundspeed so at that point I then performed full go around maneuver. We [requested priority handling] and once above the clouds I directed FO to the airspeed unreliable checklist. I suspected that icing was the culprit and as the FO ran the checklist we concurred that our airspeeds were accurate and it was likely an indication problem with our speed warning systems due to the icing. Both of our alternates had been reported as overcast (OVC); so we decided to reattempt an approach to ZZZ and requested vectors again for the ILS XXL. This time; I disengaged both the autothrottle/autopilot for the approach and asked the FO to monitor all airspeeds and call out anything less than 160kts(vref was 148). I flew the approach maintaining a higher than normal speed to stay above the min maneuver band. Since the Airspeed (AS) unreliable checklist didn't exactly reflect our situation; I decided that during the approach any stall warning indications would be respected. At some point during the approach we got the 'Leading Edge (LE) FLAPS TRANSIT' light on. As far as I know; we never exceeded any flap speeds or aircraft limits during the event. We landed; left the flaps down and had Airport Rescue and Firefighting inspect the aircraft; particularly our flight controls because we still had transit light. Everything appeared normal and we proceeded to our gate. Postflight inspection revealed some rime ice on the Leading Edge Devices (LEDs) and Maintenance discovered icing in the slat tracks. Cause: Icing. We had covered it in our brief and selected engine anti-ice but I never considered turning on the wing anti-ice and in retrospect realize that it may have mitigated some of our issues. Wing anti-ice is used so rarely that I tend to focus on the restrictions/limitations over the reasons to use it. Not anymore! Time. After [requesting priority handling] I felt time constrained because in the back of my mind I was concerned about our plan B. I knew the ceiling/vis should not be a problem getting into ZZZ; but was not sure how far we might need to go if we couldn't safely get below the OVC layer and needed an alternate with no ice on the arrival. In the rush to get back into the approach I did not give the Flight Attendants (FAs) a precautionary landing brief. I also opted to continue the approach with the transit light on because we knew it was icing related and that would take several minutes of checklist. Checklist. The airspeed unreliable checklist was our initial response to the indications; however the checklist does not consider an event that presents initially as unreliable airspeed but then once checklist is completed we still encounter PFD speed warnings anomalies. I would recommend a decision tree that considers if all airspeed indications are accurate; that icing accumulation may be a factor and direct accordingly with winganti-ice and speed additives for approach.