On the basis of her income and the information you supply above, it would intially appear to me that her income is too high to qualify for a medical card. The weekly income limit for a single person under 66 is €184, so even taking weekly rent and travel expenses from her salary, she still may be above the limit.
It may however be worth applying for a GP Visit Card. The weekly income limit there is €276, so by taking her weekly rent and travel expenses into account, she may qualify. "
The above figures refer to a single person, which she is not, since she is cohabiting. Therefore, they will be assessed as a couple and both incomes will be taken into account. So based on an assessment of income, either both will qualify or neither will qualify.
However, the HSE, as was said, does have discretion to award a medical card or GP visit card based on medical need or hardship, but you must first apply in the normal way and if you are refused, you can appeal for your girlfriend only, based on her medical need. You will need a letter detailing her illness and the costs involved and a decision will be made based on the cost of her doctor visits/medication/hospital admissions versus how much above the limit your combined income actually is.