B737-900 Captain reported Amber Leading Edge Flap Transit Light remained illuminated and Flaps had stopped close to F1 but still between 0-1 on approach. The flight crew continued to destination airport and completed a safe landing.

Narrative

Aircraft X; ZZZ1 to ZZZ; Captain (CA) was PM; First Officer (FO) was PF. While on the ZZZZZ Arrival Crossed ZZZZZ [fix] 210 Knots and 6;000 feet. ATC directed to 3;000 feet; then HDG 070. As PM I selected Flaps 1 and then HDG 070; 170 KIAS and F5 as I extended course for ILS XXR. Noted that the Amber Leading Edge Flap Transit Light remained illuminated and Flaps had stopped close to F1 but still between 0-1. Began QRH Leading Edge Flap Transit light as the initial condition. After identifying QRH Ck list with PF; I realized that I needed more time to complete the QRH and might have to complete a Flaps up landing if we were not able to provide F15 through alt flap extension procedures so [requested priority handling] provided reason as flap malfunction with souls on board and fuel remaining and requested ILS XXL for ZZZ longest runway. Next Requested long vectors and a turn out at present altitude. ATC sent us north toward ZZZ2 to de conflict with vectors and a climb to 4;000 feet. After completing initial QRH steps for Leading Edge Flap Transit Light; Flap condition 0-1 no asymmetry and then followed Ck list to its conclusion in the Alternate Flap Extension procedure. When on a long final manually set flaps to F15 using alt flap ext procedure and reviewed F15 landing data I had requested from dispatch. Once the dual response landing checklist was completed and briefing finished; broadcast to ATC that we planned a normal F15 landing; no further assistance; repeated same to ZZZ tower when asked and reiterated a normal F15 landing no further assistance. Repeated same after uneventful landing and taxi to gate. Called Dispatch; Operations and made maintenance write up for the condition.Being familiar with the long and specific QRH for Leading Edge Flap Transit light and it's possible outcomes; I chose to [advise ATC] to receive both priority handling; time and the longest runway in the event of Flaps UP landing. Knowing I could possibly achieve a normal landing and in the best case. Glad it all went according to QRH best case...and we had time to review and execute the correct Ck list and follow the complicated decision tree without outside pressure.