I think the first thing to mention is that it is Never a good idea to trek entirely alone in Nepal purely from a safety point of view, illness and small accidents can easily happen and if you don't have at least some backup these small problems can soon become Big ones !!
It is usually quite easy to find a trek-mate to go with, or of you prefer, just hire a Porter/Guide.
For a route – With such a limited timeframe, close to Kathmandu your best option might be getting a bus up to Nagarkot, overnight there then the next day trek to Chisapani, overnight there then the next day trek to Sundarijal which is on the outskirts of Kathmandu
This map should help put the above into prospective
Good Luck and Happy Safe Trekking
My Nepal Blog members.virtualtourist.com/m/50192/6ec/
This describes a similar trek but in the opposite direction:
Nagarkot Chisopani. From Chisopani instaed of the direct trail to Sungarijal you can continue in the forest by the top of Shivapuri to Buddhanilkanta, or even go as far as Kakani.
Nepal is quite popular trekking and cultural tours. you can spend many days in Nepal by exploring world heritage site, natural beauty and many things. If you are looking for 2 nights 3 days trekking around Ktm, this might be good option.
Day 01: Drive to Sundarijal and trek to Chisapani (2165m) via Borlang Bhanjyang (2430m) overnight at hotel
Day 02: Trek to Nagarkot (2195m), this is the attraction place of this trekking. It offers long range of snow caped mountains and also very famous hill station from Kathmandu for the view of sunset and sunrise. The first day trekkers can enjoy the sunset and next day morning the sunrise from your room or rooftop of the hotel.
Day 03: Trek to Changunarayan (1435m) through Tamang village, Changunarayan temple is a holy pilgrimage site of Hinduism and drive back to Kathmandu
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