Not having used Illustrator, I would say that it >>MIGHT<< just be easier to re-create your sketch from scratch within Inventor and be done with it rather than try to save time by using an import that isn't really going to get you there.
I have recently taken a PDF, placed it on a sketch and traced the line work that was of importance to me, then placed the dimensions that were critical.
Not pretty, but it does work.
Illustrator is what the name implies, an illustration tool. That said, it will not generate line work to the tolerances needed by a tool like Inventor. Often, even importing ACAD work you will see that lines that APPEAR to be connected, aren't really.
Frustrating, I know!
- Gary
If it brings in an outline of your shape you should be able to extrude this shape into a solid, you don't need a face to do this. If it does not allow you to extrude then your path is not closed. Right click a line in the sketch and select close loop, then select all lines around in a circular path untill it is closed. No different than making a square, circle shape, etc and extruding it.
If it's really complex, you might have better luck opening the drawing into AutoCAD and running the "Boundary" command ("BO" [Enter]) to create a closed polyine. It's a lot more robust and forgiving than Inventor's "Close Loop" command, from my experience. Then you can save the resulting polyline in its own file and open that into Inventor.
Don't rely on the exported lines to be perfectly precise, though. If you need your dimensions to be exact, re-create it in Inventor.
Some lines may already be connected, if so they shld turn a different color. if already closed they will not let you select them. But it may still be best just to redo in inventor as your sketch will not be dimensioned or constrained.