If my ex-boyfriend owes me $5597, how likely is it that I could get a settlement in court?

First, just as a matter of terminology--you don't get a settlement in court, you get a judgment if  you win. A settlement is an agreement--essentially a contract--which two parties voluntarily enter into to settle, or resolve, a legal action; it is not an order or award of the court.

In terms of the likelihood of you winning in court and getting a judgment in your favor: to win, you have to show the extent of the loss and that the other  person caused it, and you also have to show that you have a legal right to repayment. The documentation you describe should show the extent and cause; the question then is can you show you have a right to be be repaid. If the money--e.g. anything he charged on your credit card--was a gift at the time it was made, he does not owe you anything. To prevail, you will have to be able to show that at the time you loaned the ex-boyfriend money or let him use your charge card, there was an agreement that it was a loan and he would have to repay you.

Since you have nothing in writing stating that he acknowledges the debt or will repay you, the issue will come down to (assuming he denies the debt) how credible or persuasive your testimony is as compared to his, bearing in mind that as the  party suing, the burden of  proof is on you. That is, if all things are equal, you lose--you have to be able to show that it is more likely than not (i.e. by a "preponderance of the evidence") that he borrowed money from you and must repay.

However, no case is certain--even if your evidence is weak, it's possible you could win. As a result, people  often settle a case--enter into a voluntary agreement to pay part of what they owe--to avoid the risk of losing everything and the cost of litigation. Therefore, if your case is credible, even if not strong, it is possible that if you file a lawsuit, you will at least be able to come to some settlement with your ex-boyfriend and recover part of what he owes you.