Well that's a very interesting question.
Obviously, you'll have to trigger the AI somehow. The thing is, the AI will be decided completely by the game rules, which you know best. I can give you a high level idea of what such an AI can be:
It's always good to divide problems like this into smaller chunks, so you can, for example, say your card AI works in turns by making actions in a loop. Each such action should be done in two steps:
1. Evaluate the AI player's condition and save it. For example if the AI player has enemies next to it's arcane tower, you might want to note that. Other things can be resources, number of cards in hand, whatever you think is necessary. Save all those things, your AI player's state, and calculate a score of how good your player is doing right now.
2. If you have a small number of actions possible, for example pick and draw one card from a hand of 8, you can go over all possible actions and see how they change your state and it's score, and simply pick the one action which maximizes your score. If you have a lot of possible actions going over all of them will take a lot of time, and won't be useful at all. Instead, you can create rules for how to handle different situations. For example, if the enemy is attacking the AI player's arcane tower, then you'd probably want to deal with that, and quick. Otherwise you can go on to less pressing issues like attacking the enemy arcane tower, etc.
Of course that is just the beginning, and you can think of a lot of ways to improve that, like saving good cards for times of need, take opportunities and the likes. If you can find another map which has a card mechanic with AI, you can see how they implemented they're own and build on it (be sure to credit them).
