I'd also suggest that it might be worth paying for a taxi.
The nearest Underground station is Hyde Park Corner but getting there involves changing lines on the Underground (which involves a bit of hassle) and then finding your way across busy traffic. It's simpler to take the Victoria Line (Southbound) for 3 stops to Green Park and walk from there (straight along Piccadilly) , but it's just over ¼ mile.
If economy is important, you could take the fairly short walk from Euston Station to Euston Square Station, which is well-signposted. That's on the corner of Gower Street. Just a short way along Gower Street you'll find the terminus for the No 14 bus, which stops opposite the RAF Club on Piccadilly.
Chris