The impact would have changed the tilt of the Earth's rotation. But the impact might not have been a direct hit, but more of a glancing impact. And, moreover, the Earth would have had some angular momentum before the impact. The result is likely to have been a planet that was spinning at some angle other than exactly 90 degrees to the plane of the ecliptic.
After that, the angle can and does change. The Earth is not perfectly spherical and so gravity from the sun, moon, and even other planets can gently pull on the Earth and vary the tilt.
For the Earth, we are lucky to have the moon! It stabilises the variation and keeps it within a few degrees of 23.5° (between 22.1° and 24.5°, over a cycle of about 41000 years). Mars (which also has axial tilt despite not suffering such an impact) has probably experienced significant periods in which the axial tilt grows to as much as 90 degrees.
So while the impact did change the tilt of the Earth, it is not the cause of the current axial tilt, which is the result of more gentle gravitational perterbations. The axial tilt is not constant, though thanks to the moon it doesn't vary much. The Earth does behave like a gyroscope. But a gyroscope that is tilted will tend to preserve that tilt. It doesn't remember where it was before. And the Earth is a gyroscope in a complex gravitation field.