I can't memorize and repeat quickly the days of the week even though I studied English for several years (I always need to count with my fingers). How are the name of the days taught to children? Is there a song, a doggerel or something else?
Community wiki answer to compile the suggestions.
The Days of the Week - MapleLeafHashima
Days of the Week (Adams Family) - Michelle Lebowe
Seven Days of the Week (I Never Go to Work) - They Might Be Giants
Happy Days Theme Song - Charles Fox (score) and Norman Gimbel (lyrics)
Monday's Child Poem - Nursery Rhyme
Get 7 index cards. On each index card, write down one day of the week. If possible, write a sentence that uses the day of the week in a specific context, preferably one personally meaningful to you. Ex: On Sundays people go to church. Monday is the first day of the work week. Tuesday is when I watch -- on TV. Wednesday is hump day (the middle day of the work week). Friday is the last day of the work week. Saturday I watch football on TV. Practice the index cards.
Read English-language news, as days of the week are commonly encountered there.
Physical action can sometimes help you remember. My husband learned some Polish words because his Physical Education teacher (who was Polish) would count out the student's push-ups and other exercises using Polish numbers, week day names, month names, etc. and he remembered them quite well.
If you take a vitamin every day, a weekly pill box might help you remember the days.
To answer the question... Children are taught the days of the week by exposure, not songs or mnemonics. There may be repetition, much as I learned how to count to 20 in Spanish or all of my names (I have three middle names), but it's mostly just repetition.
I know this is less-helpful than the other suggestions made, but I hope it explains why there aren't any really catchy jingles to help. Good luck!