Everything is in the manual and also a separate BLOG from Ashley. It's unfortunate to have wasted so much time.
When importing from Tiled, Construct 2 currently only supports a subset of its features. What it does support are:
Orthogonal (square) maps
Source image with optional margin, spacing and a single transparent color
Multiple layers of tiles (note a single Tilemap object represents a single layer of tiles, so when importing you're asked which layer to import)
TMX files using XML, CSV, base64, or zlib-compressed base64 data