B737-800
Flight Crew Air Transport Pilot (ATP)
ATC Issue All Types; Aircraft Equipment Problem Less Severe; Inflight Event / Encounter Fuel Issue
Flight Crew Landed in Emergency Condition
Aircraft
During the arrival; we were being vectored from the arrival to the final approach course to runway XY when trying to deploy the flaps; they would not move.  The flap gauge showed split flaps.  The CA and I noticed after climb out from [departure airport] that the flaps were up and split.  The indicator showed up on one of the gauges and the other showed that it was further up than up position (counter clockwise past the up position).  I didn't experience any kind of asymmetry on takeoff; climb out or through level off to include the trim check.  Once we realized that the flaps would not deploy; we [notified ATC].  The CA started working the checklist items and the ATC approach controller gave us vectors until we were ready to start the approach. Once the checklist items were complete; the approached briefed up; the CA took the controls and became the PF and I reverted to the PNF/PM.  The final approach controller requested more than once for us to slow to 170 KIAS - however; the manual says to fly 40 flap Vref of flaps 40 plus an additional 40 knots which ended up being above 170 KIAS - we added a few knots for the gusty winds and flew about 179 KIAS.  As such; we ended up going around on short final because of the preceding aircraft was still on the runway.  We requested and the tower granted runway heading to 4000 feet.  Being vectored around for the next approach; being concerned of our fuel state with the LE slats stuck in the full extended position; I gave an update of our fuel of 5;000 LBS and when informing the approach controller we were ready for our next approach; they made a comment that finals were going out to 20 miles to which I replied if he was aware [of our urgent situation].  The CA made a comment to the ATC controller as well concerning the spacing as we did not want to go around again because of our faster approach speed.  I did see 19.X on the ILS DME readout on the ND when being turned base.  Upon rollout; the right main fuel tank was amber and read 1.91; the right 2.02.  Flying another approach with extended vectors would have been interesting.We landed uneventfully and was able to taxi right to our gate without incident.  ARFF followed and after clearing off; we thanked them for their assistance.  Shortly after completing the shutdown checklists; 2 of the gentlemen came up and we talked for a bit.  Explaining what the issue was and we both talked about our first approach attempt.  Without knowing the cause of the split flaps; I cannot really say.  As for being asked by the approach controller; being asked to fly slower than we were capable of due to a controllability issue; I don't think I really need to go into that.  I do understand that ATC is busy; but giving a small bubble like they did may need to be looked at.  Especially since it caused a go-around.  The other thing I feel should be of concerned is that after a go-around and [current aircraft condition]; a 20 mile final isn't what I'd call the most appropriate.
B737-800 FO reported they were unable to extend flaps upon arrival. Approach and landing were safely executed.