General aviation pilot reported the lack of a published runway closure NOTAM at LFK Airport resulted in a go-around and diversion.

Narrative

Day 0 routine operations out of LFK. Departed at night XC:25. Checked notams the day prior and saw a notam for Runway 7/25 closed unless given prior clearance 30 minutes prior to usage. Called that number and was given permission to operate as requested since the night flying did not affect their operations. There was a second notam showing a closure for Runway 16/34 for aircraft larger than 70ft. Aircraft X has a shorter wingspan so this notam did not apply. After checking the notams early in the day I saw no new developments. After getting to the aircraft I proceeded as normal (peflight; startup; sensor startup; check AWOS). The AWOS briefed the weather and only had a remark about pressure altitude. Taxied and took off 16 (winds 170@9) and noticed the lights weren't on. Attempted to turn them on and they did not illuminate so I proceeded as usual. After departing I received word from a pilot that there was a runway closure of 16/34 via notam which we believe was issued around XA:30 but no pilots recall seeing it during their pre flight preparations. Continued as normal and conducted the mission. After about 3 hours of flight time we decided to make our way back to the airport due to low fuel. (Normal flight range is about 7 hours in air but since LFK had an inoperative self fuel pump and their fuel truck is almost; if not empty we had to use the remaining fuel from the previous pilots flight giving us about 4 hours of fuel) We communicated with Houston Center that we were finished with work and heading back to LFK full stop and were met with no objections. Once airport was in sight we cancelled flight following an proceeded to fly the pattern for 25. On short final at about 100-400agl I saw what I had believed to be a dull yellow 'x' over the numbers. I decided to hold altitude and verify over the top and confirmed that there was an 'x' over the numbers. We went around and I started a climb to 2000 MSL and got back on with Houston Center. I inquired about the runway closures of 16/34 and 7/25 since that would mean our destination airport had no open runways. They confirmed that there were no known notams for the closure and called the supervisor to verify. Supervisor confirmed and called the airport to which there was no answer. After talking about it with Houston and we decided to divert to ZZZ which was a short flight but with the little amount of fuel left there was an enormous amount of pressure to land asap (As Soon As Possible). Landed ZZZ at XF:50 and had about 45 minutes of fuel remaining. Main points: pilot unknowingly took off of a closed runway; airport had no usable fuel sources and closed Runway (7/25) had no notams for closure which forced a go around and diversion with minimal fuel. Corrective actions: planning to call airport personnel in the morning to discuss the events of the night prior to hopefully prevent other traffic from making the same mistakes and personally relearned the value of checking all pre flight information immediately prior to flight as opposed to hours before and assuming no change.