Replacing
JFK-LAX-DFW-MIA-YYZ-LHR = 9,620 mi
with
JFK-SJU-LAX-DFW-YVR-LHR = 12,695 mi

gives you an additional 3,000+ miles

Replacing
SIN-BKK-BOM-HKG-NRT-SYD = 12,107 mi
with
SIN-NRT-CGK-NRT-SYD = 16,909 mi

give you an additional 4,800 miles and saves a segment.

Since your original plan was only 15 segments, you can now do an LAX-SAL-LAX (2319 mi each way)

So you end up with:
LHR-SIN-NRT-CGK-NRT-SYD-PER-CBR-SYD-JFK-SJU-LAX-SAL-LAX-DFW-YVR-LHR
For a total of 53630 miles. There might be a few more 1,000s of miles that I have optimized.

BTW, if you are buying a DONE4 ticket, starting in Poland will save you over GBP 400, but is likely to cost you in miles. Starting in South Africa will save you almost GBP 900 and is likely to get you more miles, but there is an issue of positioning yourself.