Some learned, wise people say that Swami Desikan composed the Sri Kingruhesha Stuti on Tiruvallur Emperuman. Not only is this not found in Desika Stotramala, but it also not found in any Deiska Grantham.
All can recite Sri Kingruhesha Stuti.
Notes:
Many granthams that Swami Desikan composed are lost to us. Many details are beyond our small intellect. And there are many doubts about many other details. No harm in this and let us leave them to remain as doubts.