Extract lines at the edge of the object:
img = Import["https://i.stack.imgur.com/pT8aP.jpg"]
lines = ImageLines[EdgeDetect[FillingTransform[Binarize[img]]]];
HighlightImage[img, {Thick, Yellow, Line /@ lines}]

From here you can rotate the image if you get the angle:
θ = Mean[ArcTan @@@ Subtract @@@ lines]
(* 1.67222 *)
ImageTransformation[img, RotationTransform[\[Theta] - Pi/2],
Padding -> 0, PlotRange -> All]

and from here proceed how you would have if it wasn't tilted.
On the other hand, you can try the following as well. Here I take random points from one of the lines and find the minimum distance to the other line. I then take the mean of all of these minimum distances.
pts = RandomPoint[Line[lines[[2]]], 500];
Mean[RegionDistance[Line[lines[[1]]]] /@ pts]
(* 83.175 *)
in microns:
Mean[RegionDistance[Line[lines[[1]]]] /@ pts]*Quantity[6.7, "Microns"]
(* Quantity[556.521, "Microns"] *)