Tower Controller working intersecting runways reported questionable separation when an arrival was permitted to land when the departure had yet to clear the intersection.

Narrative

While working Tower Local Control (combined with Flight Data/Clearance Delivery/Ground Control/Tower CIC); I believe I lost standard runway separation between a PA28 (on a VFR touch-and-go Runway XX) and a C172 (VFR landing Runway YY).The sun was low in the western sky with low-level haze and the Tower window shades were down due to the sun. Tower windows and shades are dusty. Visibility from the Tower was somewhat diminished by the haze and dusty shades/windows; and a support post in the northeast corner of the Tower cab obscuring part of the Runway YY final. The PA28 took much longer than expected (more like stop-and-go than touch-and-go) to begin takeoff roll after completing touch and go on Runway XX; and it took longer than normal for me to realize that. Once I realized the PA28 had taken longer than expected to roll; I looked to verify the position of the C172; which also took longer than expected due to the haze and dust and reflection of the sun on the shades. By the time I realized the PA28 was closer than I expected; they were over the Runway YY threshold and committed to land. The PA28 was airborne on runway heading after departing Runway XX and nearly through the intersection with Runway YY. There was not adequate time to send the C172 around (putting them closer to the PA28 in the air) or turn the PA28 prior to Runway YY. From the Tower vantage point; it appeared that the PA28; although airborne and at or near pattern altitude; did not quite clear the Runway YY intersection before the C172 landed on Runway YY. [I recommend] raising the northeast Tower shades or clear the PA28 to land only. (Changing the C172 to Runway XX or giving them a 360 would have made them a conflict with an E135 on final to Runway YY behind the C172.)