CRJ-200 Captain reported landing safely after experiencing a flap failure on final approach.

Narrative

Flight was progressing normally until we were cleared for a visual approach. Since it was a visual approach; I decided to fly the approach with the autopilot off. After intercepting the localizer and extending the flaps to 20 degrees; the landing gear was selected to the down position. Per the profile; I called for 'flaps 30; 160' and the First Officer (FO) moved the flaps lever to the 30 degrees position. An instant later; a FLAPS FAIL Master Caution message appeared. I continued to maintain aircraft control and started to analyze the situation. We observed that the flaps had failed in the 30 degree position. I asked the FO to [advise ATC we needed assistance] and I called for the FLAPS FAIL Quick Reference Handbook (QRH). Then I made the mistake of electing to continue when I should have elected a missed approach. The FO started to go over the FLAPS FAIL checklist in the QRH while I was following the glide slope.At that point; I realized that we would not have enough time to complete the checklist but I was wrongly focused on the fact that we were aligned with a runway that was 10;900 ft. long and on glide path. I interrupted the QRH checklist and asked the FO to calculate the landing distance and V ref for 'flaps 30' configuration. He quickly gave me the answer; the landing distance was only 20% higher than the normal distance and we had roughly 3 times more runway than the priority landing distance required and we were required to fly only 8 knots faster than a normal approach. I made the mistake of thinking that landing was the safer and wiser choice. Rather than correctly performing the missed approach; completing the checklist; and contacting Dispatch; we continued the approach to a safe touchdown. I realize I should have elected the missed approach to give us time to perform the checklist and discuss the matter with Dispatch.The landing was smooth and I deployed the one working reverser. The rollout appeared to be very reasonable and I didn't have to use much braking; I retracted the reverser by 60 kts. I stopped way before the end of the runway with light to moderate braking. As a result; all the brakes stayed in the green (BTMS 5 or less even at the jet bridge). The rollout was as smooth as a normal landing and the brakes were not hot so; at this point I decided that the [need for special handling was over]. I contacted the Flight Attendants (FA) and let him know that no further danger was apparent. Also; I told the FA that the rest of the taxi will be uneventful. The taxi to the gate and the deplaning of the passengers were uneventful. In retrospect; I should have abandoned the approach and engaged the autopilot at the first indication of the FLAPS FAIL Caution message. That would have given me the time necessary to run and complete the FLAPS FAIL QRH checklist and notify Dispatch; the FA and the passengers of the [situation].