How many observations does it take to determine the orbit of a TNO
The main problem with determining an object's orbit is we only know the position with certainty in two dimensions.  The distance to the object is largely unknown.  This accounts for the large uncertainty in the period of newly discovered TNOs.  Many possible orbits could fit the early observations, and therefore the uncertainty is large.

As time goes on, and more observations are made, more and more of these possible orbits are excluded and the orbit fit becomes more exact.  

So to answer your question, it really depends on how exact you want the answer, and how good your observations are.  The more observations you take, and the better they are, the closer your orbital fit becomes.  And more importantly, the bigger the arc of the orbit your observations cover, the more accurate the orbit becomes. It is impossible to give a specific number.

To give a practical example, Pluto takes 248 years to orbit the Sun.  It was discovered in 1930, so we've only observed it for 85 years, or about 1/3rd of one orbit.  One of the problems the New Horizons probe sent to Pluto had to overcome was that we didn't know, at launch, exactly how far away Pluto was, so therefore we didn't know exactly when New Horizons would pass the dwarf planet. That made it difficult to tell the probe where to point to take pictures at closest approach.