Does Peter mention Jesus' death, burial, resurrection, or shed blood when explaining the lame man's healing?
No. In explaining the lame man's healing, Peter attributes the miracle to faith in Jesus' Name rather than to His death, burial, resurrection, or shed blood. The text states Peter does not even allude to those elements there; instead he calls them to believe in His Name, implying that Jesus is the Christ.
Les explains that Peter and the other eleven preached the same message. In the healing explanation, Peter declares that the man was made strong through faith in Jesus' Name: "His Name, through faith in His Name, hath made this man strong." Les notes that Peter does not make reference to Jesus' death, burial, or resurrection in this healing explanation, nor does he mention the power of His shed blood—he does not even allude to it. Instead, Peter calls them to believe in His Name, and the Name Jesus Christ implies He was the Christ. Though Peter elsewhere in the sermon speaks of their killing the Prince of Life and God raising Him, the healing itself is tied to faith in His Name.