I leave the trailer connected so I'm sure it won't move.
Remove the spare from the back of the trailer. I carry a wrench that I know will fit.
Loosen the lug nuts while the trailer is still on the ground so the wheel won't turn.
Jack the trailer under its frame near the tire, not the
axle. What looks like the
axle is really a housing for the rubber suspension parts and it will be damaged by lifting there.
Some of my tow vehicles over the years have come with a jack that would work with the trailer. I now carry a scissors jack for the trailer. Use leveling blocks under it if needed for more lift.
Change the tire and tighten the lug nuts finger tight.
Gently lower the jack until the tire is touching the ground and snug the nuts.
Put everything away and use the trailer sink to clean up.
You might want to try it in your driveway sometime.