Low level sine to pulse converter with sharp rising-edges
I would not bother trying a "quick fix". I would instead try the following circuit







simulate this circuit &ndash; Schematic created using CircuitLab

Basically, it requires the addition of 1 op amp and two diodes. I've shown a TL081 as the op amp, and this will work in simulation down to 1 mV or less, but it eventually runs into input offset voltage problems. If you can find an op amp with much less input offset, you can do even better. For a real circuit, get an op amp which includes an offset adjust pin, and use this to produce an op amp output of zero with zero input.

You'll also note that the comparator output is inverted with respect to the input, but that is easily remedied in logic.

EDITED TO ADD - And in the name of all that's holy, get rid of the 1N2804. It's a 50-watt zener, and is operating far below its recommended current level. A 1N4735, as shown, has a 6.2 volt rating and appropriate current levels.