We went on to do some sight-singing, at which point I began to realize, she really doesn't have much easy materials (strange, that!) because she kept pointing to something then we would realize it's just too hard for me. What we did manage to find, I didn't manage as well as I would have done at home. There just seems to be a blank. So now I realize I need to super-duper practice at home to have a chance of performing under exam conditions. Also I've ordered some more materials which I will bring along next time.
I think you need to be a bit more diagnostic about your sight singing gaps.
Sight singing is not so much a continuum from easy to hard, it is a collection of different difficulties, each of which must be mastered individually, or a different strategy used.
Did you have trouble with tonal music? That's a different problem from missing an interval of a fifth. If you missed an interval of a fifth, is it because you failed to recognize it on the page? or because you knew it but couldn't sing it?
I would write down everything you failed at at the lesson, and target them one at a time. If you didn't sing a fifth, sing fifths all day up and down the keyboard until they are rock solid and you can never miss a fifth. Then next week do the same with a fourth. Sing them until they are perfect - no don't, you know how I feel about "perfect" intervals! (when I work on a vocal part, I play the piano part one eighth note behind while I sing in tempo. That's forced choice with immediate feedback)
If you were rhythmically imprecise, dig out the Robert Starer book or the Navy manual and your metronome, and clap rhythms every day.
Build your repertoire of things you can
And sing inner parts in church - alto and tenor take more concentration to find the notes. If you don't go to church, steal a hymnal (no sin involved, you're not a believer) and sing along with some MIDI. All hymns are online in MIDI. Hymns are tonal and that won't help you with pure leaps, but some sightsinging IS tonal and those will be easy.