I don't think Philip II Augustus would let Ingeborg marry John Lackland. If he obtains a divorce, it would be better for the both of them (Philip would have less problems with the Papacy and Ingeborg wouldn't suffer) but he would not let her marry one of his ennemies. Not to mention this could remind the French Barons of a previous similar situation : the marriage of Louis VII (who was Philip II's father...) and Eleanor of Aquitaine, who ended up with a divorce and Eleanor marrying Henry II of England.
In my opinion, if Ingeborg divorces from Philip II, she probably is sent back to Denmark with her dowry (I think it was something like 100,000 silver marks or something like that) and is never heard of again in Western Europe and may be married to a Nordic Nobles (i.e. Danish, Swedish, Norwegian or even North German prince, duke or count).
Besides that, I don't see how much this changed things. In my eyes, John Lackland is, without a doubt, not a match for Philip Augustus : Philip is on a far higher competence level than John. Having John marry Ingeborg while she had been Philip's wife would probably not change the military situation, but may increase Philip's hostility towards John. That's all I can say.
The marriage could have a butterfly though : if John is already married before he meets Isabelle of Angoulême, he won't marry her (meaning she marries Hugh X of Lusignan earlier) and this might lead Philip to search for another reason to attack John's holdings in France.