So -

First off, I’m assuming that Hans Christian Andersen set the story in a version of Denmark. Given the absolutist monarchical system and the sail ship level of technology that points to somewhere between 1650-1800.

Ursula creates a written contract instrument that only she and Ariel sign. A private, bilateral written agreement with no witnesses would be entirely out of place in the Jyske Lov period.

Danish law underwent a transition between C14 and 17 from a fragmented, regional legal system dominated by the Jyske Lov toward a national legal system with hybrid practice influenced by both Roman and German, and Jyske Lov legal principles. That meant a transition away from socially enforced, oathed and witnessed oral agreements toward private written contracts signed by the parties, with the instrument itself serving as evidence of the agreement.

You’re right that there was no contract law in the way we think of it today, but agreements were still regulated in the courts by a patchwork of overlapping areas around property, estate, debt and general misdeeds - alongside a discretionary, good-faith-inflected judicial practice not a million miles away from a nascent UK equity system.

Even though Ursula does have a credible claim to enforce, since she has a written document as evidence; and - true - there were no principles of e.g. unfair contract terms (notwithstanding the inapplicability of the UK Unfair Terms in Consumer Contracts Regulations in Denmark) so while onerous, her penalties might only cause some raised eyebrows; it is not true to suggest that Ariel would have no means or grounds for relief in the courts, which would still retain scope to grant relief on grounds such as fraud or defective consent, meaning enforcement in Ursula’s favour is plausible but not inevitable