How do you implement compile-time string conversion functions?

Proposal n4121 looks like it's going to add a std::string_literal type. It contains code like:

template<size_t n> struct string_literal { char data [n]; }

and:

template <size_t N> 
constexpr int stoi( const string_literal<N>& str, 
                    size_t* idx = 0, int base = 10); 

And here's my Number class:

template <typename T>
struct Number
{
private:
    T t;
public:
    constexpr Number(const T& t)
        : t(t)
    {
    }

    constexpr Number(const std::string& s);

    constexpr operator T() const
    {
        return t;
    }
};

template <>
constexpr Number<int>::Number(const std::string& s)
    : t(std::stoi(s))
{
}

I've taken a look at How do I convert a C string to a int at compile time? but it only works for C strings. and c_str() is non-constexpr. Also on top of that, this doesn't cover stol, stoul, stoll, stof, stod... and so on. Who knows if this proposal will make it into the standard. Also I don't want to wait 3 years for this library change to happen. How do I implement it in the moment, now?

Here's my attempt so far:

namespace lib
{
    constexpr bool is_digit(char c) {
        return c <= '9' && c >= '0';
    }

    constexpr int stoi_impl(const char* str, int value = 0) {
        return *str ?
                is_digit(*str) ?
                    stoi_impl(str + 1, (*str - '0') + value * 10)
                    : throw "compile-time-error: not a digit"
                : value;
    }

    constexpr int stoi(const char* str) {
        return stoi_impl(str);
    }

    template<size_t n> struct string_literal { char data [n]; };

    template < class charT, size_t N> 
    constexpr string_literal<N> 
        make_string_literal( const charT(&arr)[N])
        {
            string_literal<N> sl;
            for (std::size_t i = 0; i < N; ++i)
                sl.data[i] = arr[i];
            return sl;
        }
}

template <typename T>
struct Number
{
private:
    T t;
public:
    constexpr Number(const T& t)
        : t(t)
    {
    }

    constexpr Number(const std::size_t N, const lib::string_literal<N>& s);

    constexpr operator T() const
    {
        return t;
    }
};

template <>
constexpr Number<int>::Number(const std::size_t N, const lib::string_literal<N>& s)
    : t(lib::stoi(s.data))
{
}

int main()
{
    constexpr auto s = lib::make_string_literal("123456789");
    constexpr Number<int> n { sizeof(s.data), s };

    return 0;
}

main.cpp:44:69: error: non-type template argument is not a constant expression
    constexpr Number(const std::size_t N, const lib::string_literal<N>& s);
                                                                    ^
main.cpp:53:78: error: non-type template argument is not a constant expression
constexpr Number<int>::Number(const std::size_t N, const lib::string_literal<N>& s)
                                                                             ^
main.cpp:29:38: error: cannot initialize an array element of type 'char' with an lvalue of type 'const char [10]'
            return string_literal<N>{arr};
                                     ^~~
main.cpp:60:19: note: in instantiation of function template specialization 'lib::make_string_literal<char, 10>' requested here
    auto s = lib::make_string_literal("123456789");