Not sure why the thresholds... they aren't quite right.
The outputs need pull-ups, because the comparators are open collector type.
The bottom 393 has negative feedback. It may oscillate on the switching edge. LM393 is a little bit faster than the 555, so it may be fine anyway (certainly without the feedback resistors), or it may make one or two extra pulses during the edge.
If the frequency is quite low, simply using a logic inverter will do. 74HC04 is fine at 5V. Use a CMOS 555 and you can run the whole thing from +5V without worrying about dividers between stages.
If you don't mind discrete solutions, this is accurate within 20ns or so between rising and falling edges:
A couple values would have to be changed to run from a 555 and at 5V.
If you don't mind buying ICs, a CD4047 will work on 5 or 9V, and does precisely what you want, out of the box.
Tim