Just found out house is leasehold, not freehold.

I had a similar situation when I bought my house. It only came out that it was leasehold and not freehold when I'd sunk all that I had spare into fees - and it had approx 970 years remaining. I decided to proceed with buying it and over 7 years on, I can't see a single way in which it has been different than if it were freehold.
My local council are the freeholders. Not heard from them in relation to the house whatsoever (apart from me paying council tax, obviously) since I moved in.
Peppercorn rent has been zero.
Some houses in my street are leasehold and some are freehold, but the prices they sell for don't seem to be any different, so I don't think the value is really an issue in this case, with such a long lease and no chance of ground rent increase. Suppose I'll find out if I ever try to sell!

My only slight concern is that due to my house not meeting the criteria for freehold, it will never be anything other than leasehold - but obviously, I'm not going to be around when the lease comes to an end!

I purchased an ex local authority house in 2010. It was leasehold - 987 years remaining on original 999 year lease. Peppercorn rent.
The terms on the lease were that I needed to maintain the property, keep the inside in good order and redecorate at least every four years. I also needed to seek permission from the freeholder to make external changes such as building an extension or adding a porch or conservatory (a fee would be charged for this).
I have never had to pay any ground rent - I would happily have given the local council a few peppercorns if they send a demand for them!
In 2015 I decided to purchase the freehold. It cost £1548 in total including legal fees for both parties.
It hasn't made the slightest bit of difference TBH, the valuation is still the same as the neighbouring leasehold properties but at least I don't have to seek permission to make changes. Also, i'm sure it will have more appeal when I come to sell it as so many scary stories are circulating about leases that some prospective buyers would possibly have been put off.

I'd recommend the following:

1. fire your solicitor. How can he be unable to explain peppercorn rent?
2. understand whether the rent can be increased, when, how, by how much, subject to what, etc; especially should the freehold be sold to someone else
3. understand if any kind of fee is due to the freeholder for insurance, for maintenance of the driveway, or what. The difference between a 999-year lease and a freehold is not the duration, but the fact that leaseholders often have no control if the fleeceholder decides to overcharge
4. be absolutely sure that you won't need the freeholder's permission (which typically requires a fee) for extensions, changes to the windows, etc. Have you ever heard of the Dulwich Estate in South London? Google it. If you buy a freehold house in that area, they can fine you if they don't like how you keep your garden, you must ask for their permission for any changes (double glazing is a big no-no), etc. Yes, even if you own the freehold - it's not a typo.