The angle at the apex will be 45 degrees.

I alluded to the existence of a second possible solution. An interesting fact is that if you consider the orthocentre of any triangle along with the three vertices, you can in fact choose any three of those four points to form a triangle, and the resulting nine point circle will be the same, with the nine points in all the same places (although midpoints might become ‘halfway to orthocentre’ points and vice versa). So this means there are four possible triangles with the exact same set of eight points as above. However, since we are specifically asked for an isosceles triangle, there is only one additional triangle which fits the bill, ie the triangle formed by the two lower points of the above triangle, along with what was the orthocentre. This has an apex angle of 135 degrees.