What is the proper procedure for washing the towel, the purificator and other linens and vessels used in the Mass? Thank you for your help.

And thank you for asking.  This is an important subject.  Briefly, the “proper procedures” is the ‘traditional care” outlined in “Redemptionis Sacramentum” (2004, no. 120):  "Let Pastors take care that the linens for the sacred table, especially those which will receive the sacred species, are always kept clean and that they are washed in the traditional way. It is praiseworthy for this to be done by pouring the water from the first washing, done by hand, into the church's sacrarium or into the ground in a suitable place. After this a second washing can be done in the usual way."  This indication is a further specification of what was stated in the General Instruction of the Roman Missal, no. 334.
 
The basic principle is “sancta, sanctae tractandae” (holy things should be treated with reverence.)

Sacred linens – especially the purificator and the corporal – come in direct contact with the precious body and blood of Christ. When they are laundered – washed, dried, folded, ironed – they should receive the exquisite care that a person in love would give to their loved one. 
 
Would a mother who had to wipe up the blood of her son killed in a car accident, just toss that cloth in the washer with the rest of the laundry?  I don’t think so.  She might never wash it.  In the same way, we should treat the linens that touch the sacred body and blood of Christ.