Put together your IKEA BESTA frames and attach them (as instructed in the IKEA instructions) to become one, three-section unit. (Don't add the drawer inserts yet.)
Build your plywood frame. This makes the piece look more finished, and is also FAR easier to paint than the IKEA material is, so it minimizes that. Use a nail gun or drill and wood screws to attach the three pieces of plywood, with the longest piece sitting atop the other two as shown. We had our handyman do this for us and he used a nail gun. Ensure that it fits snuggly over your IKEA BESTA unit.
Sand and paint your plywood frame, sanding between coats. Let dry.
Set the frame on top of your BESTA unit and attach it. It should be flush with the back of the unit and extend over the front to cover the edge of the drawer fronts (once they are attached at the end). Use your two wood screws to attach the BESTA unit to the plywood as shown above (one on each side).
Remove the center of your HANVIKEN drawer fronts. The center piece that you are removing is very thin so it won't be hard to remove. Use a mallet or sharp tool to punch through it and carefully remove the pieces (which are simply inserted into the outside frame).
Add a wood strip to the shorter sides of the openings using wood glue. (I didn't cut mine perfectly, so I used a staple gun to hold this in place as well. But wood glue should work if you cut to the right size!) This will be used to attach your cane webbing to.
Prime the front and back of the drawer frame and let dry. Sand.
Paint the drawer frame (front and back), sanding between coats.
Cut and soak the cane webbing. Cut a piece of webbing to the size of the opening in your drawer front. Put it in a sink or bucket of warm water for 30 minutes to soak. This will cause the webbing to become more pliable and easy to work with, and it will also enlarge slightly.
Attach webbing to your drawer frame. You are going to insert the webbing into the slit where the center piece of your drawer front was removed from. NOTE: Your cane WILL be larger than the opening at this point. We will cut it down in a bit.To insert, run a line of glue along the top slit and one side of the drawer frame and insert the cane. Hold in place for 30-60 seconds until set.
Trim the cane webbing so that you can insert the other two sides into the slits in the drawer frame. Glue inside the slides and insert the webbing, hold for 30-60 seconds until set.
Use a staple gun to staple the cane webbing along the edges. (The staples will get covered with the reed next) Now, let the webbing dry completely. The webbing will NOT be perfectly taught until it is dry, so don't fret. Once it starts drying, it will tighten up and look great.
Attach the reeds. Cut pieces of reed to be the exact size of the inner frame. Cut the reed on a 45 degree angle using sharp scissors so that the ends fit together perfectly once glued on to the drawer. Use your clear adhesive to glue to the drawer, covering the staples. Let dry.
Attach the drawer fronts to the drawers and build according to IKEA instructions. You didn't change anything other than the look of the drawer front so you can follow the IKEA instructions to finish building your drawers (or cabinets) and attaching them.
Add desired hardware, handles or drawer pulls.
Add legs per instructions that come with them. Tada! Your rattan IKEA BESTA hack is complete.