B737 Captain reported a flap malfunction during final approach. Flight crew performed a flaps up landing.

Narrative

CA - PM. FO - PF. On approach into ZZZ; turning on final for the ILS XXR at 4;000ft MSL; ATC instructed us to slow to 210 knots. Slowing below 240 knots; flaps 1 was called for and the flap lever was moved to the flaps 1 position. A few moments later the LE FLAPS TRANSIT light remained illuminated and there was a difference between the left and right flap indicator needle tips. I confirmed the flap handle was in the correct flaps 1 position and the flap indicator needle tips were not aligned and not indicating flaps 1. I announced to ATC that we would need to cancel the approach; remain at 4;000ft MSL; and requested vectors so we could run appropriate checklists. The FO remained as PF and I proceeded to run series of non-normal checklists. I first ran the (NG) Flaps Leading Edge Flaps Transit non-normal checklist for the associated LE FLAPS TRANSIT light. Once complete with all applicable steps I moved on to the Flaps - Trailing Edge Flap Disagree non-normal checklist. We had a un-commanded roll and the flap needle tips had a split of one needle width. This condition lead to the (NG) Flaps - Trailing Edge Flaps Asymmetry non-normal checklist. Once complete; this lead to the (NG) Flaps - Trailing Edge Flaps Up Landing non-normal checklist. Before completing the Deferred items of the Trailing Edge Flaps Up Landing non-normal checklist; I requested priority handling with ATC and I briefed the Flight Attendants; gave them a brief and made a PA to the passengers. Once all non-normal checklists were complete; we were then cleared for the ILS XYC were an uneventful landing occurred. We exited the runway at taxiway 1 and held short of runway XYR so ARFF (Airport Rescue Fire Fighting) could approach the aircraft to ensure there was no visible damage; leaks; smoke on the aircraft. ARFF found no leaks; smoke or visible damage and advised the aircraft looked safe to continue taxi. We conducted a normal taxi to gate XX with ARFF following out of an abundance of caution. Once parked appropriate checklists and reports were completed.