The problem is that there is no day or night in Flatland - there is all-pervasive sourceless illumination. And sourceless presumably means that it is occurring inside a Flatlander as well as outside, e.g. inside the Flatlander's eye. Theoretically this ought to drown out any light coming from the outside, unless there is some mechanism (such as a lens) to concentrate a lot of external light. Even then the optic nerves would be firing constantly, making it very difficult to perceive details. We know (because Abbot tells us) that they can see, although not very well - however, their visual problems are partly a result of the 2D environment and partly caused by the constant mist that he describes, which limits visibility to a few feet.
So I think that what I need is a neural mechanism that is very good at detecting light coming in horizontally but ignores all other sources. And that is only one molecule thick (one atom theoretically, but I'm going to allow myself a little leeway here.)
My best though here is an arc of something like flat optical fibres, set up so that any photons coming in anything other than horizontally whizz out again. With something to reduce horizontal scatter, e.g. a pinhole camera mechanism. This would explain the mouth/eye structures I put on my previous drawings quite well - the lips (more like mandibles)/eyelids are opaque, with a teeny gap at the corner which works like a pinhole camera. When a Flatlander wants to eat it opens its mouth/eye and the mandibles slice off bits of food. It is somehow pushed out of the cavity and down into the rest of the flatlander as past as possible, but even so a Flatlander will effectively be blind for a few seconds every time it eats a mouth-full of food.
One result of this is that the narrower figures (especially isosceles and females) will have very narrow fields of vision, which could be a problem.
Okay, that's my first thoughts - anyone able to suggest a better mechanism?