B737-800 Captain reported a flap disagree during approach. The flight crew executed a missed approach; were able to set the flaps using an alternate method; and continued to a normal landing.

Narrative

On approach to ZZZ flight #X. Cleared ILS approach Runway XXR. Captured ILS and started configuring the aircraft to flaps10; for an ATC assigned speed of 170 KIAS. The aircraft was not slowing as I anticipated. I asked the copilot to select flaps 10. He responded the flap lever was at 10. I checked the trailing edge flap gauge and cycled the flap handle to 5 then back to 10. No change! Flaps were remaining at flaps 5. I instructed my copilot to report a missed approach.I climbed the aircraft straight ahead and complied with ATC instructions. I placed the autopilot on and transferred Pilot Flying duties and communications with ATC to my copilot. I opened the QRH and accomplished the trailing edge flaps disagree checklist. I [requested priority handling]; requested Runway XXL (the longest runway) for approach and landing. I requested and received non normal landing data via ACARS and notified ZZZ operations of our situation.I notified my flight attendants of the problem using the briefing format; advising a landing in 10 minutes. I addressed the passengers via the PA of our problem; and assured them of a successful outcome.At approximately 15 miles out on the ILS Runway XXL; I completed the checklist items of configuring the flaps to 15 via the alternate method and completed the dual response landing checklist. At 10 miles on the ILS I assumed the role of Pilot flying and landed.Landing and rollout was normal. We continued to the gate and advised maintenance. After all passengers had deplaned; I debriefed with the crew and thanked each of them for their excellent work.