T. danae is a known, and probably important, component of the diet of the sperm whale (eg, (Clarke, 1967; Okutani & Satake, 1978). T. danae can be swallowed whole to at least 32 cm ML (T. K., personal observations) but larger ones, judging from the following account, must be torn up before swallowing.
Underwater observations and photographs of sperm whales feeding on squid are rare. According to a report written by the observers Wade and Robyn Hughes, Wayne and Pam Osborn, and Tania Windsor Blunden (personal communication): A young male sperm whale, about six to seven meters in length, surfaced off Faial in the Azores with a large squid in its jaws. Initially the whale "appeared to have ... taken [the squid] head on, with [the squid's] swimming fins hanging symmetrically down each side of the whale’s jaws." "After swimming on or near the surface for about ten minutes with the squid held in its jaws, the whale sank slowly on its side, and then on its back convulsively chewing and snatching at the squid’s carcass." In the picture below, on the right, fragments of the squid float near the whale indicating the "sloppy feeding" seen by the observers. They estimated the squid's width from fin tip to fin tip as 1.5 to 2 m; this width would approximate the squid's mantle length.
Figure. Sperm whale eating a squid, probably Taningia danae, photographed off the islalnd of Faial in the Azores. Left - Recently surfaced sperm whale and squid. Right - Later photograph in which the whale has a diffenent grasp of the squid and fragments of the squid are floating nearby. Photographs taken by Wade and Robyn Hughes while free diving (i.e., without scuba gear).