I am from NI, and I would say 'Ash-leen'. However, if you actually pay attention to the NI accent, you realise that many people would pronounce 'ing' as 'een' in many cases. 'Sitting' would sound like 'sitteen', 'walking' like 'walkeen'. 'Sing' or 'ring' would be pronounced with a short 'i' sound and a recognisable 'g' sound, but the gerund would (in many accents, including my own) sound like 'singeen' or 'ringeen'.
I find this discussion quite weird for that reason. Aisling is pronounced 'Ashling', but how that sounds is a whole other issue.