Shop for and purchase a twin bed connector. You can likely find one at a bed and bath store or specialty bedding store. They are also available from online bedding retailers. Several variations of this product are available. Some are simply a long strip of foam to fill the gap between the two mattresses, while others include a strap that goes around the perimeter of the mattresses to hold them together. Still others are pads that cover the entire surface of both mattresses.
Slide your trundle bed out from under your standard bed and raise the trundle until the two mattresses are the same height. Remove any sheets or blankets on the mattresses.
Place the twin bed connector on the two mattresses. Depending on the mattress connector you purchased, this step can be as simple as positioning a strip of foam along the gap between the beds or as involved as strapping the two mattresses together. Follow the manufacturer's instructions for your mattress connector. Although the two mattresses are pushed together until they touch, there will be a small gap or indentation where they meet. The mattress connector fills in and disguises the gap.
Place a king-size mattress pad and king-size fitted sheet over the two mattresses. This will further mask the gap and help keep the two mattresses from shifting apart.
Test to see how securely your mattresses are held together by lying down in the middle of the bed where the two twin mattress edges meet. If the mattresses are not strapped together around their perimeter and shift apart, purchase a strap-type mattress connector, which has more holding power. If you do have a strap-type connector and the mattresses shift, tighten the strap to better secure the mattresses.