Electrical – How to deal with strobe and balance pins of this comparator for single supply operation

You need a flip-flop and a reset button. You don't need anything fancy like a latch or master-slave flip-flop (called flip-flop in 7400 terms). Simply 2 NAND or NOR gates will do.

This is a slightly modified circuit used in many copies of an original circuit from AADE. The designer (the late Neil Heckt) was generous (or naïve) enough to publish all the details including the equations used internally so the design was widely copied, usually without credit, and never with any licensing fees.

Anyway, as others have pointed out, the short to Vcc should be replaced by a resistor, maybe 1K. The LC oscillator operates at the resonant frequency of L1/C7, so:

\$L_1 = \frac{1}{C_7(2\pi f)^2}\$

The original oscillator (from here):

The Oscillator

The key to L/C Meter IIB's operation is the oscillator circuit of FIGURE 1. The LM311 is a voltage comparator. When power is applied, the voltage at pin 2 is 2.5 volts causing the output to be at a level of 5 volts. This charges capacitor C4 through resistor R4 until the voltage at pin 3 equals 2.5 volts. As it reaches 2.5 volts the output switches to a low level inducing a transient into the tank circuit composed of L1 and C1. The transient causes the turned circuit to ring at it's resonant frequency. The ringing causes a square wave at the resonant frequency to appear at the output of the voltage comparitor. The square wave is coupled back to the tuned circuit through R3 and C3 sustaining oscillation.

For the nominal values of L1 (68 uH) and C1 (680 pF) an increase in L of 1 nH (.001 uH) or an increase in C of .01 pF produces a frequency change of slightly more than 5 Hz. A 0.2 second measuring period can resolve 5 Hz and therefore .001 uH or .01 pF.

I don't think the order of magnitude increase in hysteresis from using a 10K resistor for R5 in your circuit is an improvement. It may well cause other problems. Nor is reducing the RC product of R4/C4 an improvement.