Ampere Turn in Electromagnet

You can't. You haven't got the pertinent information- the number of turns as well as the current.

Also, for DC, 220V, 13.5A doesn't correspond to a resistance of 13.7 ohms.

Amp-turns means amps times turns. So you have to know the number of turns. You can only get this physically by somehow counting the turns or estimating the turns by measuring the coil dimensions and then measuring the gauge of wire used to wind the coil and from that and the resistance find the length of wire and hence the approximate number of turns in the coil. However the resistance of copper wire is notoriously variable and the packing of the wire in the coil plays a role in accuracy.

How about this: Wind a coil of known turns (n2) over the magnet winding, apply AC voltage (v1) to the magnet coil, measure the voltage (v2) appearing across the added winding.

v1 / n1 = v2 / n2

magnet turns = test winding turns * ( applied volts / test winding volts )

Yeah, sort of works. Big problem is the actual coupling to the electromagnet may not be accurately known. If the electromagnet has an iron core coupling would be good and if you put an iron "keeper" on it it would be even better and the method would work well. If it's just a big fat coil of wire, getting the mutual coupling high will be harder to do. Still would be a good first approximation. Obviously the most accurate way is to count the turns when you build the coil (or look them up in the coil specs if it was commercial)