An appeal can only be withdrawn by the claimant or their authorised representative by giving notice in writing – Reg. 29(9) of the Decisions & Appeals Regs.
If the solicitor in this case did not have signed authorisation to act on the claimant’s behalf, then the appeal has not been validly withdrawn, and still stands.
Therefore there is no need for the claimant to make a fresh, and late, appeal.
I presume the appeal hadn’t been forwarded to the Appeals Service in the 16 or so weeks betwen the original appeal and the letter from the solicitor, which, if that is the case, is a bit naughty. If it had, then similar rules apply, under Reg. 40 of the Social Security and Child Support (Decisions and Appeals) Regs.
If the appeal had been validly withdrawn, then it cannot be reinstated, but the claimant can make a fresh, albeit late, appeal.
In the circumstances you describe, therefore, unless there is evidence that the solicitor was authorised to act on the claimant’s behalf, the original appeal is still valid.