Yet another example of why a shareholders' agreement should always be entered into - in fact it should be made compulsory where the shares are held 50/50!
But as I assume there isn't one the obvious solution is for you to buy her shares.
Without a shareholders' agreement there is no simple way of forcing her to sell them to you, but you at least need to talk to her and find out if she's willing in principle to sell.
If she is then it's just a matter of agreeing a price. I note that you're divorced, in which case I'd have expected a formal valuation of the shares to have been obtained within the divorce proceedings, or at least for an agreed value to have been placed on the shares, in which case you can use that valuation as the basis for your offer.
If, for any reason, no valuation has been obtained / agreed, and you can't reach agreement, then you will need to obtain a formal valuation.
As you're going to be the continuing director it would almost certainly be beneficial to have your existing accountants provide the valuation. They will naturally be `on your side', as they will be expecting repeat business. You can justify choosing them to your ex by pointing out that they are familiar with the company, so will be a lot cheaper than external accountants.
However, assuming she doesn't fall for that you will need to agree to employ an independent firm of accountants to value the shares.
If she just refuses to co-operate at all then you should obtain your own back of the envelope valuation and use it to make her a formal settlement offer, known as a Part 36 Offer. This will effectively force her to get legal advice, as she won't understand what the offer is, which should get things moving.
If it doesn't, then probably the best remedy is to issue a Winding Up Petition on the basis that the company is a quasi-partnership and that because of the deadlock it can't function.
In practice, the court has the power to order her to sell you her shares at a price to be fixed by the court, but it's very unlikely it would ever get to a hearing, as merely issuing the Petition should be enough to bring her to her senses and to the negotiating table.
All the above assumes, of course, that there is cash available to finance the share purchase. If you don't have the cash then it may be possible for the company to use its own resources to buy the shares itself, though it's quite complicated, and you would need appropriate legal advice.
