You must define that her activity is a business.
Educational institutions that are in business
These establishments require payment for the education they provide, so their education is a business activity:
Education and training is exempt from VAT for business.
An educational institution is exempt.
The fees for the instruction that are paid are exempt from tax. She is not a non profit sporting service.
This is instruction her activity is involved in.
Under the same VAT notice you cited the conditions are spelled out.
Her business is instructing not a sporting activity.
They are paid under PAYE so their amounts are not subject to VAT. Your wife's business is the VAT subject not employees.
The amounts charged to students is under the business not individual instructors.
Please do not forget or ignore to give a positive rating (look for the STARS or SMILEY FACES). It adds nothing to your costs but it assists me.