I suspected that he saw the number .999883 and thought, six significant digits. I wanted him to know that that value only has three significant digits.
If the question is, how did I get even a three significant digit measurement, that's easy to do with good DVM's using the method I described to him (and which you are recommending).
On the other hand, trying to get more than about 1 significant digit by some method involving leakage inductance will be difficult with a mains frequency iron core transformer.
But if you ask me if I believe the k value of .999883 is *accurate* to 3 significant digits, that's another question. The OP didn't ask that; he just asked how I got "such a precise number", and I wanted to be sure that he understood that .999883 is not *precise* to 6 digits. :-)
Trying to get repeatable measurements from a mains frequency iron core transformer is not easy. I find that if I just try to measure the self-inductance of a winding at 60 Hz and some excitation level, the reading will drift for many minutes, sometimes taking 5 minutes or more before the measurement is stable to 3 digits. Apparently the initial transient of connecting the meter tweaks the core and it takes a while to relax, and if the transformer has just been connected to line power, it can take even longer!
I mentioned that flux density at the excitation level of the measurement, temperature and magnetic history of the core could affect measured k. How much would depend on the particular transformer, of course.
The 1943 book I refer to in the earlier thread says it well (about a method for measuring leakage inductance):
"...this method is inherently inaccurate when used with *measured* values of the self- and mutual inductances of iron-core transformers. The leakage inductance of one winding of such a transformer often may be as small as
0.2 per cent of its self-inductance. For example, if the self-inductance of winding 1 is 10 henries, its leakage inductance may be about 0.02 henry. If the value of the leakage inductance is to be determined from Eq. 91, to the nearest millihenry--or within about 5 per cent of its true value--the value of the self-inductance must measured to the nearest millihenry, or within 0.01 per cent of its true value, and the mutual inductance must be measured with the same per cent accuracy. Such precise measurements are impossible with iron-core transformers."