Peter Kibisu was lodging with trainee lawyer Elizabeth Nnyanzi and her family at their Harrow home when he attacked the 31-year-old in her bedroom.
The court heard Kibisu, 24, had tried to cover his tracks after the murder in August 2015 but was caught when police found his DNA on the victim.
He was ordered to serve a minimum jail term of 27 years at the Old Bailey.
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Judge Richard Marks QC said Ms Nnyanzi was an "outstanding young woman of immense talent and ability" who had died as a result of "unspeakable wickedness".
Mark Heywood QC told the court Kibisu had been living with the Nnyanzi family for nine months and was regarded as a younger "cousin" by the victim.
On 14 August the 24-year-old returned to the house after a drink and drug-fuelled night out to find Ms Nnyanzi alone, and raped and killed her.
He tried to cover his tracks by sending a text message to the victim while on the way to work later that day.
When he returned he called police to say he had discovered the victim in her room not breathing and the home's rear patio doors ajar.
Kibisu was arrested after a pair of bloodied boxer shorts were found in a bin and a tissue sample Ms Nnyanzi's fingernails matched his DNA.
Det Ch Insp Tim Duffield said the 24-year-old only admitted his guilt "when confronted with a whole raft of evidence".
In a statement Coleen Nnyanzi, the victim's mother, called Kibisu a "wolf in sheep's clothing" who had turned on a family that had "helped him in his hour of need."

Vice News's Jake Hanrahan and Philip Pendlebury, and Iraqi Mohammed Ismael Rasool, were detained in Diyarbakir, south-east Turkey on Thursday.
Vice News said they had been moved to a "high security prison" five hours away.
Police reportedly questioned them about alleged links to Islamic State and Kurdish militants, charges they deny.
Vice News has condemned the charges as "ridiculous", and a "baseless" attempt by the Turkish authorities to censor their coverage.
The group had been filming clashes between police and youths from the outlawed Kurdistan Workers' Party (PKK) which have erupted in recent weeks, the organisation said.
Turkey has limited journalists' access to the region.
A Foreign Office spokesman said it was "concerned" about the arrests, and expected Turkey to uphold EU obligations to protect press freedom.
"Respect for freedom of expression and the right of media to operate without restriction are fundamental in any democratic society," he said.
As a state part to the European Convention on Human Rights and UN Declaration of Human Rights, "we would expect the Turkish authorities to uphold the obligations enshrined in those agreements", he said.
Consular officials were in contact with the group, he added.
A lawyer for the group told the BBC they had been detained by police in their hotel, and their cameras and computers had been confiscated.
They were formally charged by a Turkish court with "working on behalf of a terrorist organisation" on Monday. They have denied all charges.
They have until next week to lodge an appeal, but their lawyer said he was "not optimistic" about securing their release.
Kevin Sutcliffe, Vice's head of news programming in Europe, said they had now been transported to a prison facility more than five hours away from where their legal representation was based, and from the court where they are due to appear.
"This move appears to be a blatant obstruction of the fair legal process that Turkey has repeatedly pledged to uphold," he said.
Vice News said all three men were experienced reporters and had worked extensively abroad.
Mr Hanrahan, from Northampton, and Mr Pendlebury, from Wigan, had worked together on numerous stories, including the migrant crisis in Calais, it said.
Journalist and translator Mr Rasool, who was born in Iraq and is based in Turkey, has worked extensively in the Middle East with the Associated Press, Anatolia Agency, and Al Jazeera, it said.
No further court dates have yet been set.
A mechanic who admitted raping and strangling a family friend who he was living with has been jailed for life.
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The Foreign Office has said it is "concerned" about the arrest of two UK journalists and their fixer in Turkey, accused of helping a terrorist group.