At what time were you travelling? First class or standard? I am a little puzzled as to how the travelling ticket inspector has arrived at £185.
Indeed, that's the first thing I'd be wondering. I, too, can't see any obvious way that a particular fare currently in the system would lead to a Penalty Fare of £185, as there are only four walk-up single tickets from Leicester to St Pancras - the Super Off-Peak and Anytime Singles, for Standard and First Class. Twice any of them makes for exact figures of £127, £170, £179, or £222 for a Penalty Fare, depending on the time of travel. Of course it's possible the OP has rounded or approximated the amount of the Penalty Fare, but if the Penalty Fare has been issued for an incorrect amount, this is a viable ground for appeal.
A further viable ground for appeal could be the lack of signage at Leicester station complying with the required Regulations - see
here for more information. As far as I'm aware, (and I've travelled with almost every TOC!), the only TOC that has Regulations-compliant signage is SWR, and even there only at some stations.
To the OP: it is a strict liability (i.e. intent doesn't matter, like speeding) offence, under Railway Byelaw 18(1), to board a train without a valid ticket on you. In the context of a mobile ticket, it's not clear what 'on [your] person' (from the Byelaw) exactly means, and whether or not an unactivated ticket still counts as 'a ticket'. However, Railway Byelaw 18(2) requires you to show a ticket on demand, so therefore even if Byelaw 18(1) is not relevant, this probably is.
Either way - what I am pointing out is that failing to have a valid ticket is a serious matter, and a Penalty Fare is a method of regularising such an irregularity. I would therefore strongly advise paying it in the first instance. This is because, once a Penalty Fare has been paid, and an appeal submitted against it, you can
no longer be prosecuted if the appeals body makes a response to the appeal, or the deadline for them to do so passes. In other words, even if you later manage to get your money back, a Penalty Fare is a way of stopping this from becoming a criminal matter, which is always a good thing.
Of course, if this is all down to an app fault of some kind, rather than a failure to activate the ticket prior to boarding, then you may be able to hold the retailer liable for any costs you incur, including a Penalty Fare, and you would be under no obligation to appeal against such a Penalty Fare to recover its cost.
But what will
not be a valid ground of appeal is that the situation 'seems' unfair, as the title of the thread says.
One question I'd be wondering is how the OP got through the ticket barriers at Leicester - or, indeed, did they board elsewhere? If they were given permission to board the train without an activated ticket, that could be a further ground of appeal, though of course proving this is always difficult.