The SLC seem to be a bunch of crooks.
My wife has had 1 payment left of £160 for about 3 years. And has regularly phoned up/written to them asking to pay this final payment. They finally got in touch on the 27th of June to say they were going to take payment from her next salary, so she phoned them the next day to pay off the final amount which she did, and they said they would issue a "stop notice" on the account.
Fast forward 5 days (yes less than a weeks notice), they took a percentage from her salary which was significantly larger than the £160,
I would agree that they're not the most helpful to deal with usually.
What your wife could have done (I would agree that it's not made super clear and isn't helpful now) was work out when she comes to within 1 year of paying off her student loan by automatic payment and informed Student Finance that she'll be making manual payments in her last year. This means that they could have issued a STOP notice then (4 years ago). They do allow this and it prevents issues where they take more than they should.
Five days is unfortunately too optimistic to expect a STOP notice to take effect. The process is, as I understand it: Student Loans Company issue a STOP notice to HMRC. HMRC informs your wife's company's payroll department and they stop taking student loan deductions through PAYE. This means that Student Finance or HMRC could have been slow, or your wife's company's payroll department got the notice too late for it to take effect in the next payroll run. If you inform HMRC and/or Student Loans Company that your wife overpaid, I think you can get a refund from HMRC or Student Loans Company.