Yes you should have to only pay the £10 once - if you get any hassle, polite reply that you are trying to sort out the debt and a) you are unable to do this without the statements, and b) should it ever go legal, then the courts would not look too kindly on the cc company for not giving you the opportunity to settle the debt out of court.
In order to do much with the account, you will need your partner to give her permission to allow you to act on her behalf. As this is a special contract, ie she is giving you something (permission to act on her bahalf) and you are giving her nothing, her signature will probably need to be witnessed by someone else. This is almost like a power of attorney arrangement.
Are we talking the original debt arising from 1985, or expenditure since that date as well? If all the debts are from 1985, I think we may have quite an interesting issue here. You should ask the company to list what the underlying debts relate to, i.e. the original purchases. If they are unable to provide this information, and your partner has done nothing in the last 6 years to acknowledge the debt, then you may be able to get out this without paying a penny. Even if the company have continued to send statements. then I think they still need to be able to provide evidence as to what the debt is made up from. B/fwd from 1990 plus lots of interest does not represent adequate proof that your partner ever purchased anything!
Legally, I believe, your partner needs to have acknowledged the debt in the last 6 years (ie made a payment) or the bank needs to be able to prove that they have been sending statements and they can give details of what makes up the current total.