Anything less than 60 minutes would be looked at by the airline, anything less than the legally required amount (30 minutes for a jet of this type iirc) will result in a very serious investigation. Note that for slower aircraft (for instance a turbo-prop) the time requirement goes up not down because they may have to spend more time in the air to reach an alternate (or secondary alternate, if things are really bad, like what happened here).

They should investigate after the first failed landing, regardless of the amount of fuel in the tank.

Go arounds are perfectly normal and are not a 'failed landing', a failed landing is a crash.