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I was thinking of glueing the first few pieces together to make a 12 inch square piece and then drilling and hand nailing it to the floor.
Sounds reasonable. You could use a little flooring adhesive there as well and just try to blind nail the flooring.
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Also, when I transition from the miters to the apron, what would be the best way to secure it all together?
This is where one runs the floor boards long, then trims them off with a circular saw and straight edge. This must be done very carefully, especially if your flooring is prefinished. After trimming, then you route a groove all around that 45ed field. Since you are installing bamboo, you will need to make your own splines (slip tongues) that match the groove of the sides of the flooring. Then you glue the spline into the groove you just routed and then fit the first course of your apron into the spline. Basically, what you've done is made a tongue around the complete perimeter of your 45ed field floor ready to accept the groove of your apron boards. Then just build the apron one course at a time till you get to the walls. Blind/toe nail as much as you can. You may need to hand nail the last few rows and face nail the last one or two. You can also use some flooring adhesive on those face-nailed last rows to cut down on the amount of face nailing.