22 December 2015 Last updated at 11:22 GMT
The two male suspects are believed to have scaled a fence in the back garden and broken into the house in Ashbourne Road, Mitcham, sometime between Saturday morning and Monday afternoon when the theft was discovered.
The thieves damaged cuddly toys, puzzles and games belonging to the victim's' baby and stole gifts including jewellery, photographic equipment, a tablet and a laptop computer.

The cuts affected power stations and public transport, including Istanbul's tram and metro systems.
A crisis centre has been set up at the energy ministry.
Prime Minister Ahmet Davutoglu has said all possible causes are being examined, including terrorism. The cut hit at 10:36 (07:36 GMT) on Tuesday morning.
Power cuts were reported in at least 40 of Turkey's 81 provinces.
Traffic lights stopped working in several parts of Istanbul and Ankara during the outage, causing traffic jams, while rescue teams were called to help people trapped in lifts or stuck underground at metro stations.
By Tuesday afternoon, electricity returned to parts of Istanbul. Energy Minister Taner Yildiz said about "80% of Istanbul's electricity has been restored" and he expected power to return to the rest of the country soon.
Istanbul's transport company said all of its routes were back up and running.
Turkish TV also reported that power had been re-established in the cities of Edirne, Tekirdag, Erzurum and Trabzon.
"Our main target right now is to restore the network. This is not an incident that we see frequently," Mr Yildiz told Turkish media.
Turkey suffers from sporadic electricity cuts but locals say they cannot remember such a nationwide cut for a generation, reports the BBC's Mark Lowen in Istanbul.
One of the few cities unaffected by the power cut was Van in the east of the country, where electricity is supplied from Iran.
CCTV footage has been released of thieves stealing gifts from under a Christmas tree at a home in south London.
[NEXT_CONCEPT]
A massive power cut has hit dozens of provinces across Turkey, with officials saying a break in connections with mainland Europe could be to blame.